Orbital Speed

An object in orbit is constantly being pulled toward the body it orbits. Rather than escaping gravity, an orbiting object is actually in continuous free fall.

For a circular orbit, gravity supplies exactly the inward acceleration needed to keep the object moving along its curved path.

The circular orbital speed is:

v = √(GM/r)

where:

• v = orbital speed in m/s
• G = gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
• M = mass of the central body in kg
• r = orbital radius measured from the centre of the central body in m

The equation shows that orbital speed depends on the mass of the central body and the orbital radius, but not on the mass of the satellite.

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Orbiting Objects Are Falling

Imagine throwing a ball horizontally from a high mountain.

Gravity pulls the ball downward, so it follows a curved path and eventually strikes Earth.

If the ball is thrown faster:

• it travels farther horizontally
• Earth curves away beneath it
• it takes longer to reach the surface

If it could be launched sufficiently fast and atmospheric resistance were ignored, the curvature of its falling path could match the curvature of Earth.

The ball would continue falling without reaching the surface.

It would be in orbit.

This idea is often called Newton's cannonball thought experiment.

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Continuous Free Fall

A satellite in orbit experiences gravitational acceleration.

Therefore, it is continuously falling toward Earth.

However, it also has a large sideways velocity.

During the time the satellite falls toward Earth, the curved surface of Earth falls away beneath it.

The satellite therefore continually misses Earth.

This produces an orbit.

The important idea is:

An orbit is not the absence of falling. An orbit is a particular kind of continuous falling.

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Gravity Does Not Disappear in Orbit

A common misconception is that satellites orbit because they are beyond Earth's gravity.

They are not.

Earth's gravity extends indefinitely, becoming weaker with increasing distance.

At the altitude of many artificial satellites, gravity is still quite strong.

Without Earth's gravity, a satellite would not remain in orbit.

It would move in approximately a straight line according to Newton's First Law.

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Gravity Bends the Path

At any instant, a satellite's velocity is approximately tangent to its circular orbit.

Gravity acts toward Earth's centre.

Therefore:

• velocity points sideways
• gravitational force points inward
• gravitational acceleration points inward

Gravity continuously changes the direction of the velocity.

This bends the satellite's path into a circle.

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Constant Speed but Changing Velocity

For an ideal circular orbit:

speed is constant

but:

velocity is continuously changing

This happens because velocity includes both:

• magnitude
• direction

Even if the magnitude remains constant, the direction changes at every point around the orbit.

Therefore, the satellite is accelerating.

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Centripetal Acceleration

Any object moving in a circle requires an inward acceleration called centripetal acceleration.

Its magnitude is:

a_c = v²/r

For a satellite, gravity supplies this acceleration.

Gravitational field strength at distance r is:

g = GM/r²

Therefore, for a circular orbit:

a_c = g

or:

v²/r = GM/r²

This relationship allows orbital speed to be calculated.

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Deriving Circular Orbital Speed

The gravitational force on a satellite of mass m is:

F_gravity = GMm/r²

The required centripetal force is:

F_c = mv²/r

For a circular orbit, gravity provides the centripetal force:

GMm/r² = mv²/r

Cancel the satellite mass m:

GM/r² = v²/r

Multiply by r:

GM/r = v²

Take the square root:

v = √(GM/r)

This is the equation for circular orbital speed.

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Why Satellite Mass Does Not Matter

Notice that the satellite mass m cancels during the derivation.

A larger satellite experiences a larger gravitational force:

F = GMm/r²

but it also requires proportionally more force to produce the same acceleration:

F = ma

Therefore, satellites of different masses can orbit at the same speed if they are at the same orbital radius around the same central body.

A 10 kg satellite and a 10,000 kg satellite would require the same ideal circular orbital speed at the same location.

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What Determines Orbital Speed?

From:

v = √(GM/r)

orbital speed depends on two quantities:

• mass of the central body, M
• orbital radius, r

For satellites around the same planet, M remains constant.

Therefore:

v ∝ 1/√r

This means orbital speed decreases as orbital radius increases.

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Orbital Radius

Orbital radius is measured from the centre of the planet, not from its surface.

Therefore:

r = R + h

where:

• R = radius of the planet
• h = altitude above the surface

This distinction is essential in orbital calculations.

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Worked Example: Finding Orbital Radius

A satellite is:

600 km above Earth's surface

Earth's radius is approximately:

6370 km

Therefore:

r = 6370 + 600

= 6970 km

Convert to metres:

r = 6.97 × 10⁶ m

This is the value that must be used in the orbital-speed equation.

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Calculating Orbital Speed

Suppose a satellite orbits Earth at:

r = 7.00 × 10⁶ m

Use:

G = 6.67 × 10⁻¹¹ N·m²/kg²

M_Earth = 5.97 × 10²⁴ kg

Then:

v = √(GM/r)

v = √[(6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(7.00 × 10⁶)]

v ≈ √(5.69 × 10⁷)

v ≈ 7.54 × 10³ m/s

Therefore:

v ≈ 7.5 km/s

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Low Earth Orbit

Many satellites orbit relatively close to Earth in low Earth orbit (LEO).

Typical orbital speeds are around:

7–8 km/s

That means a satellite can travel thousands of kilometres in only a few minutes.

---

Example: Satellite 400 km Above Earth

Suppose:

h = 400 km

Earth radius:

R = 6370 km

Therefore:

r = 6370 + 400

= 6770 km

= 6.77 × 10⁶ m

Now:

v = √[(6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(6.77 × 10⁶)]

v ≈ 7.67 × 10³ m/s

Therefore:

v ≈ 7.7 km/s

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Orbital Speed and Altitude

For satellites orbiting the same planet:

v ∝ 1/√r

Therefore:

higher orbit → lower orbital speed

and:

lower orbit → higher orbital speed

This may seem surprising because higher satellites have farther to travel around their larger orbits.

However, gravity is weaker at greater distances.

Less centripetal acceleration is required, so the satellite travels more slowly.

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Comparing Two Orbital Radii

Suppose Satellite A orbits at radius:

r

Satellite B orbits at:

4r

Since:

v ∝ 1/√r

then:

vB/vA = √(r/4r)

= 1/2

Therefore:

vB = 0.5vA

A satellite orbiting four times farther from the centre moves at half the circular orbital speed.

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Another Comparison

Suppose Satellite A orbits at:

r

Satellite B orbits at:

9r

Then:

vB/vA = √(1/9)

= 1/3

Satellite B moves at one-third the orbital speed of Satellite A.

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Orbital-Speed Pattern

For satellites around the same central body:

Orbital speed follows an inverse square-root relationship with radius.

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Comparing Satellites at Different Altitudes

Consider two satellites around Earth.

Satellite A is in a low orbit.

Satellite B is much farther from Earth.

Satellite A experiences:

• stronger gravity
• greater centripetal acceleration
• greater orbital speed

Satellite B experiences:

• weaker gravity
• smaller centripetal acceleration
• lower orbital speed

However, Satellite B travels around a much larger orbital path.

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Higher Orbits Also Have Longer Periods

Orbital period T is the time required to complete one orbit.

For a circular orbit:

T = 2πr/v

A higher satellite has:

• larger r
• smaller v

Both effects increase the orbital period.

Therefore:

higher orbit → longer orbital period

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Worked Example: Orbital Period

Suppose:

r = 7.0 × 10⁶ m

v = 7.5 × 10³ m/s

Then:

T = 2πr/v

= 2π(7.0 × 10⁶)/(7.5 × 10³)

≈ 5860 s

Convert to minutes:

5860/60 ≈ 97.7 min

So the satellite completes approximately one orbit every:

98 minutes

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Kepler's Third Law

Combining:

v = √(GM/r)

with:

T = 2πr/v

gives:

T = 2π√(r³/GM)

Therefore:

T² ∝ r³

for objects orbiting the same central body.

This relationship is known as Kepler's Third Law.

It shows that orbital period increases rapidly as orbital radius increases.

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Low Orbit Versus Geostationary Orbit

A low-Earth-orbit satellite may:

• travel at roughly 7–8 km/s
• complete an orbit in roughly 90–100 minutes

A geostationary satellite travels much farther from Earth.

Its altitude is approximately:

35,786 km

Its orbital speed is only about:

3.1 km/s

and its orbital period matches Earth's rotation.

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Geostationary Satellites

A geostationary satellite appears to remain above approximately the same point on Earth's equator.

To achieve this, it must:

• orbit above the equator
• move in the same direction as Earth's rotation
• have an approximately circular orbit
• have an orbital period equal to Earth's rotational period

This requires a specific orbital radius.

If the satellite were significantly closer to Earth, it would orbit too quickly.

If it were significantly farther away, it would orbit too slowly.

---

Why Higher Orbits Are Slower

We can understand this from the force equation.

Gravitational force decreases as:

1/r²

At larger r, gravity provides less inward acceleration.

Therefore, the satellite does not need to move as quickly for gravity to curve its path into the required circle.

Mathematically:

v²/r = GM/r²

so:

v² = GM/r

and:

v = √(GM/r)

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Orbital Speed Around Other Bodies

The same equation works for any approximately spherical gravitational source:

v = √(GM/r)

This includes:

• Earth
• Moon
• Mars
• Jupiter
• Sun
• other stars

Only M and r change.

---

Orbiting the Moon

Suppose a spacecraft orbits the Moon at:

r = 2.00 × 10⁶ m

Moon mass:

M = 7.35 × 10²² kg

Then:

v = √[(6.67 × 10⁻¹¹)(7.35 × 10²²)/(2.00 × 10⁶)]

v ≈ 1.57 × 10³ m/s

Therefore:

v ≈ 1.6 km/s

This is much lower than a typical low-Earth orbital speed.

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Why Lunar Orbital Speeds Are Lower

The Moon has much less mass than Earth.

Therefore, its gravitational field is weaker at comparable relative distances.

Less centripetal acceleration is required for a stable circular orbit.

This results in lower orbital speeds.

---

Earth Orbiting the Sun

Earth itself is an orbiting object.

For a simplified circular model:

v = √(GM_Sun/r)

Earth travels around the Sun at approximately:

30 km/s

This means Earth moves almost 30 kilometres through its orbit every second.

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The Same Physics Applies

The same basic relationship explains:

• artificial satellites around Earth
• the Moon around Earth
• planets around the Sun
• moons around planets
• stars orbiting within binary systems

Gravity provides the inward acceleration that continually changes the direction of motion.

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Analyzing Satellite Motion

Consider a satellite in a circular orbit.

At any instant:

velocity points tangent to the orbit

while:

gravity points toward the centre

The satellite would naturally move in a straight line because of inertia.

Gravity continually deflects it from that straight path.

The combination creates orbital motion.

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What If Gravity Suddenly Disappeared?

Imagine Earth's gravity suddenly vanished.

The satellite would no longer curve around Earth.

According to Newton's First Law, it would continue with its instantaneous velocity.

Therefore, it would move along a line tangent to its former orbit.

This demonstrates that gravity is responsible for continually bending the path.

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What If the Satellite Slows Down?

Suppose a satellite is initially in a circular orbit and suddenly loses some forward speed.

It no longer has the correct speed for that circular orbit.

Gravity bends its trajectory more strongly than required to maintain the original circle.

The satellite enters a new orbit with a lower point.

If enough speed is lost, the new trajectory may enter the atmosphere or intersect the planet.

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What If the Satellite Speeds Up?

Suppose a satellite receives a forward engine burn.

Its kinetic energy increases.

It no longer follows its original circular orbit.

Instead, it generally enters an elliptical orbit with a higher point on the opposite side.

If sufficient energy is added, it may eventually reach escape conditions.

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Circular Orbit Is a Specific Condition

At a particular radius, there is a particular speed required for an ideal circular orbit:

v = √(GM/r)

Too little speed:

the object falls into a lower trajectory.

Correct speed:

circular orbit.

Greater speed but below escape conditions:

typically an elliptical bound orbit.

At sufficient speed:

the object can escape.

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Circular and Elliptical Orbits

Not all satellite orbits are circular.

Many are elliptical.

In an elliptical orbit:

• orbital radius changes
• gravitational potential energy changes
• kinetic energy changes
• orbital speed changes

The circular orbital-speed equation cannot simply be used at every point of an elliptical orbit.

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Speed in an Elliptical Orbit

An object in an elliptical orbit moves:

fastest when closest to the central body

and:

slowest when farthest away

The closest point is called periapsis.

The farthest point is called apoapsis.

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Energy Changes in an Elliptical Orbit

Near periapsis:

• gravitational potential energy is lower
• kinetic energy is greater
• speed is greater

Near apoapsis:

• gravitational potential energy is higher
• kinetic energy is smaller
• speed is lower

Mechanical energy remains constant if no non-conservative forces do work.

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Energy in a Circular Orbit

For a circular orbit:

Eₖ = 1/2 mv²

Since:

v² = GM/r

then:

Eₖ = GMm/(2r)

Gravitational potential energy is:

Eₚ = −GMm/r

Therefore:

Eₚ = −2Eₖ

Total energy is:

E_total = Eₖ + Eₚ

so:

E_total = −GMm/(2r)

The negative total energy shows that the satellite is gravitationally bound.

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Higher Orbit and Energy

A satellite in a higher circular orbit moves more slowly.

However, its total mechanical energy is actually greater because it is less negative.

For a circular orbit:

E_total = −GMm/(2r)

As r increases:

E_total approaches zero.

Therefore, moving a satellite into a higher circular orbit requires energy even though its final orbital speed is lower.

This is one of the counterintuitive features of orbital mechanics.

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How Can Adding Energy Make the Final Speed Lower?

Suppose a spacecraft fires its engine forward.

Initially:

• speed increases
• kinetic energy increases

This places the spacecraft into an elliptical transfer orbit.

As it climbs:

• gravitational potential energy increases
• kinetic energy decreases
• speed decreases

After another manoeuvre establishes a higher circular orbit, the final orbital speed can be lower than the original low-orbit speed.

Yet the spacecraft has more total mechanical energy.

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Orbital Speed and Escape Speed

At the same radius:

v_orbit = √(GM/r)

Escape speed is:

v_escape = √(2GM/r)

Therefore:

v_escape = √2 v_orbit

Since:

√2 ≈ 1.414

escape speed is approximately:

1.414 times circular orbital speed

at the same location.

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Worked Example

A satellite has circular orbital speed:

7.2 km/s

The escape speed at that radius is:

v_escape = √2(7.2)

≈ 10.2 km/s

Therefore, the satellite is moving below the local escape speed and remains gravitationally bound.

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Astronauts and Apparent Weightlessness

Astronauts aboard an orbiting spacecraft experience apparent weightlessness.

This is not because gravity has disappeared.

The spacecraft, astronauts, and objects inside are all falling toward Earth together.

Because the floor does not provide the normal supporting force experienced when standing on Earth, astronauts appear weightless.

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Microgravity

The term microgravity is often used to describe the environment inside an orbiting spacecraft.

Gravity itself may still be substantial.

The apparent weightlessness results mainly from continuous free fall.

Small residual accelerations can arise from:

• atmospheric drag
• spacecraft rotation
• vibrations
• tidal effects
• onboard equipment

Therefore, the environment is called microgravity rather than perfectly zero gravity.

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Atmospheric Drag and Orbital Decay

Low-orbiting satellites may encounter traces of Earth's atmosphere.

Even though the atmosphere is extremely thin at these altitudes, drag can gradually remove orbital energy.

This causes orbital decay.

As the orbit becomes lower:

• atmospheric density increases
• drag increases
• energy loss becomes faster

Eventually, the satellite may re-enter the atmosphere.

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The Orbital-Decay Paradox

Drag initially slows a satellite.

Yet a satellite in a lower circular orbit has a higher orbital speed.

This seems contradictory.

The explanation involves energy.

Drag removes total mechanical energy.

The satellite moves into a lower orbit where gravitational potential energy is more negative.

Part of that decrease in potential energy appears as kinetic energy.

Therefore, the final lower circular orbit can have a higher speed even though drag caused the orbital energy loss.

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Real Satellite Motion

Real satellites are affected by more than the simple two-body gravitational model.

Their trajectories may be influenced by:

• atmospheric drag
• Earth's non-spherical shape
• the Moon
• the Sun
• solar radiation pressure
• other planets
• engine burns

Nevertheless:

v = √(GM/r)

is an extremely useful model for approximately circular orbits.

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A Strategy for Orbital-Speed Problems

First, identify the central body's mass:

M

Then determine the orbital radius:

r = R + h

Convert all distances to metres.

Use:

v = √(GM/r)

For comparisons around the same body:

v₂/v₁ = √(r₁/r₂)

For orbital period:

T = 2πr/v

For local escape speed:

v_escape = √2 v_orbit

Finally, check whether the answer makes physical sense:

higher circular orbit should normally mean lower orbital speed.

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Common Mistakes

Thinking orbiting objects are not falling

Orbiting objects are in continuous free fall.

Thinking gravity is absent in orbit

Gravity provides the centripetal acceleration required for orbit.

Thinking constant speed means zero acceleration

The direction of velocity changes continuously, so the satellite accelerates.

Using altitude instead of orbital radius

Use:

r = R + h

Thinking higher satellites move faster

For circular orbits around the same body:

higher r → lower v

Thinking satellite mass affects orbital speed

Satellite mass cancels from the orbital-speed equation.

Treating gravity and centripetal force as two separate forces

Gravity is the force providing the required centripetal force.

Using the circular-orbit equation for an elliptical orbit

Speed changes throughout an elliptical orbit.

Thinking astronauts float because gravity is zero

They experience apparent weightlessness because they and their spacecraft are falling together.

Thinking adding energy must produce a faster final circular orbit

A higher circular orbit has greater total energy but lower orbital speed.

Confusing orbital speed with escape speed

At the same radius:

v_escape = √2 v_orbit

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Key Terms

Orbital speed — magnitude of the velocity of an object moving in orbit.

Orbital velocity — vector describing both the speed and direction of an orbiting object.

Orbit — gravitationally curved trajectory around another body.

Circular orbit — orbit with constant distance from the central body.

Free fall — motion in which gravity is the dominant force acting on an object.

Continuous free fall — description of orbital motion in which an object continually falls toward a body while moving sideways fast enough to keep missing it.

Centripetal force — net inward force required for circular motion.

Centripetal acceleration — acceleration directed toward the centre of circular motion.

Tangential velocity — velocity directed tangent to the orbital path.

Orbital radius — distance from the centre of the central body to the orbiting object.

Altitude — height above the surface of a celestial body.

Orbital period — time required to complete one orbit.

Low Earth orbit (LEO) — relatively low-altitude orbit around Earth.

Geostationary orbit — equatorial orbit with a period matching Earth's rotation so the satellite appears approximately stationary over one location.

Elliptical orbit — orbit whose distance from the central body varies.

Periapsis — point in an orbit closest to the central body.

Apoapsis — point in an orbit farthest from the central body.

Orbital decay — gradual reduction in orbital energy and altitude, often caused by drag.

Microgravity — condition of very small apparent weight experienced in an orbiting spacecraft.

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Key Takeaways

• Orbiting objects remain in continuous free fall.
• A satellite does not orbit because gravity disappears; it orbits because gravity continually bends its path.
• A satellite's velocity is tangential to its orbit while gravitational acceleration points inward.
• Gravity supplies the centripetal force required for circular motion.
• Circular orbital speed is v = √(GM/r).
• Orbital speed does not depend on satellite mass.
• Orbital radius is measured from the centre of the central body.
• For a satellite above a planet, r = R + h.
• Around the same body, orbital speed varies as 1/√r.
• Lower circular orbits have higher orbital speeds.
• Higher circular orbits have lower orbital speeds.
• Higher orbits also have longer orbital periods.
• For objects around the same central body, T² ∝ r³.
• A circular orbit has constant speed but continuously changing velocity.
• An orbiting satellite therefore has centripetal acceleration even when its speed is constant.
• Astronauts experience apparent weightlessness because they and their spacecraft are falling together.
• Elliptical orbits have changing orbital speeds.
• Objects in elliptical orbits move fastest near periapsis and slowest near apoapsis.
• A higher circular orbit has lower speed but greater total mechanical energy.
• At the same radius, v_escape = √2 v_orbit.
• Changes in satellite speed alter the size and shape of its orbit.
• Orbital speed links gravity, circular motion, energy and satellite motion into a single physical model.

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Check Your Understanding

1. Explain why a satellite in orbit is described as being in continuous free fall.
2. If gravity suddenly disappeared, what path would an orbiting satellite follow? Explain your answer.
3. Explain why a satellite can have constant speed while still accelerating.
4. Starting with GMm/r² = mv²/r, derive the circular orbital-speed equation.
5. Explain why satellite mass does not appear in v = √(GM/r).
6. A satellite orbits 500 km above Earth. Earth's radius is 6370 km. Calculate its orbital radius in metres.
7. Using M_Earth = 5.97 × 10²⁴ kg, calculate the orbital speed of the satellite in Question 6. Use G = 6.67 × 10⁻¹¹ N·m²/kg².
8. Satellite A orbits at radius r with speed v. Satellite B orbits the same planet at radius 4r. Determine Satellite B's orbital speed in terms of v.
9. Another satellite orbits at 9r. Determine its orbital speed relative to a satellite at r.
10. Explain why satellites at greater orbital radii move more slowly but take longer to complete an orbit.
11. A satellite has orbital radius 8.0 × 10⁶ m and speed 7.0 × 10³ m/s. Calculate its centripetal acceleration.
12. A satellite's circular orbital speed is 6.5 km/s. Calculate the escape speed at the same radius.
13. Explain why astronauts aboard an orbiting spacecraft appear weightless even though Earth's gravitational field is still acting strongly on them.
14. Describe what happens to the trajectory of a satellite if its forward speed is suddenly increased while remaining below escape conditions.
15. A student says, "A satellite in a higher orbit must move faster because it has farther to travel." Evaluate this statement using the relationship between orbital speed, orbital radius, gravitational field strength, and orbital period.

Types of Orbits

An orbit is the curved path followed by an object as it moves under the influence of gravity. Artificial satellites can be placed into many different types of orbits depending on what the satellite is designed to do.

Two important descriptions of an orbit are its shape and its orientation.

By shape, an orbit may be:

• circular
• elliptical

By altitude and orientation, common Earth satellite orbits include:

• Low Earth Orbit (LEO)
• Medium Earth Orbit (MEO)
• Geostationary Earth Orbit (GEO)
• polar orbit
• Sun-synchronous orbit

There is no single "best" orbit. Engineers choose an orbit based on the mission objective.

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Circular Orbits

A circular orbit has a constant orbital radius.

The satellite remains the same distance from the centre of the planet throughout its orbit.

For an ideal circular orbit:

• orbital radius remains constant
• orbital speed remains constant
• gravitational potential energy remains constant
• kinetic energy remains constant
• total mechanical energy remains constant

However, velocity still changes because its direction changes continuously.

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Circular Orbital Speed

For a circular orbit:

v = √(GM/r)

where:

• v = orbital speed
• G = gravitational constant
• M = mass of the central body
• r = orbital radius

At a particular orbital radius, there is a specific speed required for an ideal circular orbit.

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Why Circular Orbits Are Useful

Circular or nearly circular orbits are useful when engineers want predictable:

• altitude
• orbital speed
• communication distance
• observation conditions
• coverage patterns

Many communication, navigation and Earth-observation satellites use approximately circular orbits.

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Elliptical Orbits

An elliptical orbit has the shape of an ellipse.

The central body lies at one focus of the ellipse rather than at its geometric centre.

The satellite's distance from the central body changes continuously.

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Periapsis and Apoapsis

The closest point of an orbit to the central body is called the:

periapsis

The farthest point is called the:

apoapsis

For an Earth orbit, these have special names:

perigee = closest point to Earth

apogee = farthest point from Earth

For an orbit around the Sun:

perihelion = closest point to the Sun

aphelion = farthest point from the Sun

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Speed Changes in an Elliptical Orbit

An object in an elliptical orbit does not travel at constant speed.

It moves:

fastest at periapsis

and:

slowest at apoapsis

This occurs because gravitational potential energy and kinetic energy continually transform into one another.

Near periapsis:

• gravitational potential energy is lower
• kinetic energy is greater
• speed is greater

Near apoapsis:

• gravitational potential energy is higher
• kinetic energy is smaller
• speed is lower

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Energy in an Elliptical Orbit

If no non-conservative forces do work:

E_total = Eₖ + Eₚ = constant

As the satellite moves inward:

Eₚ decreases → Eₖ increases

As it moves outward:

Eₖ decreases → Eₚ increases

Therefore, speed changes naturally throughout the orbit.

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Circular Versus Elliptical Orbits

A circle can actually be considered a special case of an ellipse with zero eccentricity.

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Orbital Eccentricity

The shape of an orbit can be described using eccentricity, e.

For a circular orbit:

e = 0

For an elliptical orbit:

0 < e < 1

A small eccentricity means the orbit is nearly circular.

An eccentricity closer to 1 means the ellipse is more stretched.

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Why Elliptical Orbits Are Useful

Elliptical orbits can be useful when a spacecraft needs to:

• spend more time over a particular region
• travel between different orbital altitudes
• reach another planet
• transfer between circular orbits
• perform scientific observations at different distances

Transfer trajectories between orbits are often elliptical.

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Classifying Earth Satellite Orbits

Earth satellites can also be classified according to:

• altitude
• inclination
• orbital period
• direction of travel
• relationship with Earth's rotation

Some of the most important categories are:

LEO, MEO, GEO and polar orbit.

These categories can overlap.

For example, a satellite can simultaneously be:

low-Earth + polar + Sun-synchronous + nearly circular.

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Low Earth Orbit

Low Earth Orbit (LEO) refers broadly to relatively low-altitude Earth orbits.

LEO extends from just above the atmosphere to roughly:

2000 km altitude

Many satellites operate at a few hundred kilometres above Earth.

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Characteristics of LEO

LEO satellites generally have:

• relatively low altitude
• high orbital speed
• short orbital periods
• relatively short communication distances to Earth
• relatively small coverage area at any instant

Typical orbital speeds are around:

7–8 km/s

Many complete an orbit in roughly:

90–130 minutes

depending on altitude.

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Advantages of LEO

Low Earth orbit offers several important advantages.

Because satellites are relatively close to Earth:

• images can have high spatial resolution
• communication signals travel shorter distances
• signal delay can be relatively low
• less launch energy is required than for very high Earth orbits
• smaller transmitters may be sufficient for some applications

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Disadvantages of LEO

A single LEO satellite covers only part of Earth at one time.

It also moves rapidly across the sky.

Therefore:

• continuous coverage may require many satellites
• ground stations must track moving satellites
• satellites may frequently move below the horizon
• atmospheric drag can affect lower LEO satellites
• orbital decay may occur

Large satellite constellations can overcome some of these limitations.

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Common Uses of LEO

LEO is commonly used for:

• Earth observation
• weather monitoring
• scientific research
• crewed spaceflight
• remote sensing
• some communication systems
• imaging
• environmental monitoring

The International Space Station is also in low Earth orbit.

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Medium Earth Orbit

Medium Earth Orbit (MEO) lies above LEO and below geostationary orbit.

There is a broad range of possible MEO altitudes.

One particularly important application is:

satellite navigation.

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Why Navigation Satellites Use MEO

Navigation systems need satellites to:

• cover large areas
• remain visible for substantial periods
• provide signals from several directions
• form reliable global constellations

MEO provides a useful compromise between:

• coverage
• signal strength
• orbital period
• number of satellites required

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MEO Compared with LEO

A MEO satellite is farther from Earth.

Therefore:

orbital speed is lower

and:

orbital period is longer

It can also see a larger fraction of Earth's surface at once.

However, communication distances are greater.

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Geostationary Earth Orbit

A geostationary orbit is a special circular orbit in which a satellite appears to remain above approximately the same point on Earth's equator.

Its altitude is approximately:

35,786 km above Earth's surface

Its orbital radius is approximately:

42,164 km from Earth's centre

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Requirements for a Geostationary Orbit

A geostationary satellite must:

• follow an approximately circular orbit
• orbit above Earth's equator
• travel in the same direction as Earth's rotation
• have an orbital period equal to Earth's sidereal rotation period

These conditions allow the satellite to remain above approximately the same longitude.

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Why GEO Satellites Appear Stationary

Earth rotates eastward.

A geostationary satellite also orbits eastward.

If their angular speeds match, the satellite remains approximately above the same location relative to Earth's surface.

To an observer on Earth, the satellite appears nearly stationary in the sky.

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Advantages of Geostationary Orbit

The nearly fixed position is extremely useful for communication.

A ground antenna can point toward the satellite and does not need to continuously track it.

A GEO satellite can also view a very large fraction of Earth's surface.

These properties make GEO useful for:

• communications
• television broadcasting
• weather monitoring
• telecommunications
• continuous regional observation

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Weather Satellites in GEO

A geostationary weather satellite can continuously observe the same large region.

This is valuable for tracking:

• tropical cyclones
• cloud systems
• major storms
• atmospheric motion
• developing weather patterns

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Disadvantages of GEO

Geostationary satellites are very far from Earth.

This creates several disadvantages:

• greater launch energy is required
• signals travel much farther
• communication latency is greater
• imaging resolution can be lower for a given instrument size
• coverage near the poles is poor

Therefore, GEO is not ideal for every mission.

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Polar Orbits

A polar orbit passes approximately over Earth's north and south polar regions.

Instead of orbiting around the equator, the orbital plane is highly inclined relative to Earth's equator.

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Earth Rotates Beneath a Polar Satellite

As a polar satellite travels around Earth, Earth rotates beneath its orbital path.

On the next orbit, the satellite passes over a different strip of Earth's surface.

Over many orbits, the satellite can observe almost the entire planet.

This makes polar orbits particularly useful for global observation.

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Advantages of Polar Orbits

Polar or near-polar orbits are excellent for:

• mapping
• environmental monitoring
• ice monitoring
• land-use studies
• climate research
• ocean observation
• reconnaissance
• global Earth imaging

Unlike a geostationary satellite, a polar satellite can observe high-latitude and polar regions very effectively.

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Disadvantages of Polar Orbits

A single polar satellite does not continuously remain over one location.

Instead, it repeatedly passes over different areas.

Therefore:

• continuous observation of one region is difficult
• ground stations must wait for passes
• continuous communication may require a constellation or relay network

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Sun-Synchronous Orbits

A Sun-synchronous orbit (SSO) is a special type of near-polar orbit.

Its orbital plane slowly rotates so that the satellite passes over a location at approximately the same local solar time on successive passes.

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Why Sun-Synchronous Orbits Are Useful

Suppose a satellite photographs the same city on different days.

If every image is taken at roughly the same local solar time, then:

• Sun angle is similar
• shadows are more consistent
• lighting conditions are easier to compare

This is extremely useful for studying changes over time.

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Uses of Sun-Synchronous Orbits

Sun-synchronous orbits are widely used for:

• Earth imaging
• environmental monitoring
• agriculture
• forestry
• mapping
• ice studies
• disaster monitoring
• climate observations

SSO is therefore especially important for remote-sensing satellites.

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Equatorial Orbits

An equatorial orbit lies close to Earth's equatorial plane.

Its inclination is close to:

0°

Geostationary orbit is one important example.

Equatorial orbits can be useful when missions focus on:

• equatorial regions
• communications
• low-inclination operations

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Inclined Orbits

Not every orbit is equatorial or polar.

An inclined orbit crosses the equator at an angle.

The angle between the orbital plane and Earth's equatorial plane is called the inclination.

Examples:

• 0° → equatorial orbit
• about 90° → polar orbit
• intermediate values → inclined orbit

Inclination determines which latitudes a satellite can pass over.

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Inclination and Ground Coverage

A satellite generally cannot directly pass over latitudes greater than its orbital inclination.

For example, a satellite with:

30° inclination

can pass over regions between approximately:

30° N and 30° S

A near-polar satellite can pass over almost every latitude.

Therefore, inclination is a major part of mission planning.

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Prograde and Retrograde Orbits

A prograde orbit travels in the same general direction as Earth's rotation.

A retrograde orbit travels in the opposite direction.

Most launches use prograde orbits when possible because Earth's rotation provides some initial eastward velocity.

This can reduce launch energy requirements.

Sun-synchronous satellites, however, commonly use slightly retrograde near-polar orbits to achieve the required orbital-plane precession.

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Highly Elliptical Orbits

Some satellites use highly elliptical orbits (HEO).

These have:

• low perigee
• very high apogee
• large eccentricity

Because satellites move slowly near apogee, they can spend a large fraction of each orbit over a particular region.

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High-Latitude Communications

Geostationary satellites appear low on the horizon when viewed from very high latitudes.

This can make GEO communication difficult in polar regions.

Highly elliptical orbits can provide better viewing geometry for high-latitude regions.

A satellite can spend many hours near its high-altitude apogee over the desired hemisphere.

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Molniya-Type Orbits

A well-known example of a highly elliptical orbit is a Molniya-type orbit.

Such orbits are designed so that satellites spend a large amount of time over high northern latitudes.

Several satellites can be used together to provide repeated regional coverage.

This illustrates how an elliptical orbit can be deliberately chosen because its changing orbital speed is useful.

---

Transfer Orbits

Elliptical orbits are also commonly used to move spacecraft between circular orbits.

A transfer orbit connects an initial orbit to a different orbit.

One important example is the:

Hohmann transfer orbit

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Hohmann Transfer

Suppose a spacecraft begins in a low circular orbit and needs to reach a higher circular orbit.

A simplified Hohmann transfer works like this:

1. spacecraft begins in lower circular orbit
2. engine fires forward
3. spacecraft enters an elliptical transfer orbit
4. spacecraft climbs toward the higher orbit
5. spacecraft reaches apogee
6. another engine burn establishes the higher circular orbit

The transfer ellipse touches both circular orbits.

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Geostationary Transfer Orbit

Satellites intended for geostationary orbit may first be placed into an elliptical geostationary transfer orbit (GTO).

The spacecraft then raises and circularizes its orbit near geostationary altitude.

This avoids requiring the launch vehicle to place the spacecraft directly into its final circular GEO in a single step.

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Comparing LEO, GEO and Polar Orbits

These orbit descriptions represent different characteristics.

LEO primarily describes altitude.

GEO describes a very specific combination of:

• altitude
• period
• inclination
• direction

Polar describes orbital inclination.

Therefore, LEO and polar are not mutually exclusive.

A satellite can be both:

LEO and polar

at the same time.

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Major Comparison

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Orbit Selection Depends on the Mission

Engineers do not choose an orbit simply because one orbit is "better."

They ask:

What must the satellite accomplish?

Different objectives lead to different orbital choices.

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Mission: High-Resolution Earth Imaging

Suppose a satellite must photograph Earth's surface in high detail.

A lower altitude generally allows better spatial resolution for a given imaging system.

A suitable choice might be:

LEO

often combined with:

polar or Sun-synchronous orientation

This allows detailed observations across much of Earth.

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Mission: Continuous Weather Monitoring

Suppose meteorologists want to continuously watch a large region.

A good choice is:

GEO

The satellite can remain above approximately the same region and repeatedly observe developing weather systems.

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Mission: Global Environmental Mapping

Suppose scientists want to map:

• forests
• glaciers
• oceans
• land-use changes

A good choice is often:

polar or Sun-synchronous LEO

Earth rotates beneath the satellite, allowing global coverage over time.

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Mission: Satellite Navigation

A navigation system requires:

• broad coverage
• several satellites visible simultaneously
• predictable orbital geometry

A common solution is:

MEO constellations

This provides a useful compromise between coverage and satellite number.

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Mission: High-Latitude Communications

Geostationary satellites can be difficult to use at very high latitudes because they appear close to the horizon.

A highly elliptical orbit can provide better high-latitude coverage.

The satellite spends substantial time near apogee over the target region.

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Mission: Human Spaceflight

Crewed Earth-orbiting missions often use LEO because:

• launch requirements are lower than for distant orbits
• communication delays are small
• Earth is relatively close
• emergency return is more practical
• resupply is easier than in higher orbits

This does not make LEO easy—it simply makes it more accessible than many alternatives.

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Satellite Constellations

One satellite may not provide enough coverage.

A satellite constellation consists of multiple satellites arranged in coordinated orbits.

Constellations can provide:

• continuous global communications
• navigation
• frequent Earth imaging
• weather observations
• backup if individual satellites fail

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LEO Constellations

A single LEO satellite moves quickly relative to Earth's surface.

Continuous communication therefore requires many satellites.

As one satellite moves below the horizon, the user's connection can be transferred to another satellite.

This is called a handover.

The disadvantage is that large numbers of satellites and complex ground systems may be required.

---

GEO Coverage

A GEO satellite sees a much larger portion of Earth at once.

A relatively small number of GEO satellites can provide broad coverage across much of the planet.

However, polar coverage remains poor.

The much greater signal distance also creates greater communication delay.

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Signal Travel Time

Electromagnetic signals travel at approximately:

3.0 × 10⁸ m/s

A signal sent to a satellite must travel:

• from Earth to the satellite
• often back to Earth

A GEO satellite is tens of thousands of kilometres away.

A LEO satellite may be only hundreds of kilometres away.

Therefore:

LEO generally provides lower latency than GEO.

This can matter for:

• voice communication
• video calls
• interactive internet
• remote control systems

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Orbital Period and Mission Design

Orbital period strongly affects how often a satellite passes over a location.

Short-period LEO satellites may make many revolutions each day.

GEO satellites complete approximately one orbit per sidereal day.

Highly elliptical satellites may be designed so their periods and orientations produce repeated coverage of selected regions.

Therefore, orbital period is not merely a consequence of the orbit—it is often part of the mission design.

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Changing an Orbit

A satellite can change its orbit by firing its engines.

An engine burn changes the satellite's:

velocity

which changes its:

orbital energy

and therefore its trajectory.

A forward burn can raise part of the orbit.

A backward burn can lower part of the orbit.

Further burns can circularize the new orbit.

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Orbit Raising

Suppose a satellite begins in a low circular orbit.

A forward engine burn increases its speed.

The satellite enters an ellipse whose apogee is higher.

At apogee, another forward burn can raise the perigee and create a higher circular orbit.

Although the final higher circular orbit has a lower orbital speed, energy had to be added to reach it.

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Orbit Lowering

A spacecraft can fire its engine opposite its direction of travel.

This decreases its orbital energy.

The opposite side of the orbit drops.

This principle can be used for:

• changing operational orbits
• atmospheric re-entry
• satellite disposal

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Orbital Debris and Orbit Selection

Orbit selection also involves safety.

Earth orbit contains:

• operating satellites
• inactive satellites
• spent rocket stages
• fragments from collisions or breakups

Even small objects can be dangerous because orbital speeds are several kilometres per second.

Mission planners therefore consider collision probability and debris environments when selecting and maintaining orbits.

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End-of-Life Planning

Satellites cannot simply be ignored when their missions finish.

Depending on orbit, engineers may plan to:

• lower the satellite so it re-enters
• move it to a disposal orbit
• remove remaining stored energy
• reduce the chance of creating debris

Responsible orbit selection therefore includes the satellite's entire life cycle, not just its operational phase.

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Choosing Between Orbit Types

Consider four different mission objectives.

Satellite A: Detailed Crop Imaging

Needs:

• detailed surface images
• repeated global coverage
• similar lighting

Suitable orbit:

Sun-synchronous LEO

Satellite B: Continuous Regional Weather Monitoring

Needs:

• constant view of one large region

Suitable orbit:

GEO

Satellite C: Global Navigation

Needs:

• large-area coverage
• multiple satellites visible

Suitable orbit:

MEO constellation

Satellite D: High-Latitude Communications

Needs:

• long visibility over northern regions

Possible solution:

highly elliptical orbit

The orbit follows from the mission requirements.

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Orbit Selection as an Engineering Trade-Off

Selecting an orbit usually requires balancing competing advantages.

A lower orbit may provide:

• better image resolution
• lower signal delay
• easier launch access

but also:

• smaller coverage
• shorter visibility
• greater atmospheric drag

A higher orbit may provide:

• greater coverage
• longer visibility

but also:

• greater communication distance
• higher launch requirements
• lower image resolution for a comparable instrument

No orbit maximizes every desirable property.

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A Mission-Planning Checklist

When selecting an orbit, engineers may ask:

• What area must be observed?
• Is continuous coverage required?
• What image resolution is needed?
• How often must the satellite revisit a location?
• Are polar regions important?
• How much communication delay is acceptable?
• How much launch energy is available?
• How long must the satellite operate?
• How much atmospheric drag is acceptable?
• Is a constellation required?
• What debris risks exist?
• What happens at the end of the mission?

The answers determine which orbital characteristics are most useful.

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Common Mistakes

Thinking every orbit is circular

Many natural and artificial orbits are elliptical.

Thinking the central body is at the centre of an elliptical orbit

It lies at one focus.

Thinking satellites move at constant speed in elliptical orbits

They move fastest near periapsis and slowest near apoapsis.

Thinking LEO, GEO and polar describe the same property

LEO mainly describes altitude, GEO describes a specific orbit, and polar describes inclination.

Thinking a polar satellite must be far from Earth

Many polar satellites operate in LEO.

Thinking geostationary satellites can orbit above any latitude

A true geostationary satellite must orbit above the equator.

Thinking GEO satellites provide excellent polar coverage

Their viewing geometry is poor at high latitudes.

Thinking higher satellites move faster

For circular Earth orbits, higher satellites move more slowly.

Thinking GEO means the satellite is physically stationary

It is moving at several kilometres per second; it only appears stationary relative to Earth's surface.

Thinking one orbit is best for every satellite

Orbit selection depends on mission objectives.

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Key Terms

Orbit — curved gravitational trajectory of an object around another body.

Circular orbit — orbit with constant orbital radius.

Elliptical orbit — oval-shaped orbit in which orbital radius changes.

Eccentricity — numerical measure of how much an orbit differs from a circle.

Periapsis — closest point in an orbit to the central body.

Apoapsis — farthest point in an orbit from the central body.

Perigee — closest point to Earth.

Apogee — farthest point from Earth.

Low Earth Orbit (LEO) — relatively low-altitude Earth orbit, broadly below about 2000 km.

Medium Earth Orbit (MEO) — region above LEO and below geostationary altitude.

Geostationary Earth Orbit (GEO) — circular equatorial orbit in which a satellite appears approximately stationary above one longitude.

Polar orbit — highly inclined orbit passing near Earth's poles.

Sun-synchronous orbit — near-polar orbit designed to pass locations at approximately the same local solar time.

Inclination — angle between an orbital plane and a reference plane such as Earth's equator.

Prograde orbit — orbit travelling in the same general direction as the central body's rotation.

Retrograde orbit — orbit travelling opposite the central body's rotation.

Highly elliptical orbit (HEO) — orbit with a large difference between periapsis and apoapsis.

Transfer orbit — temporary orbit used to move between other orbits.

Hohmann transfer — efficient elliptical transfer between approximately circular orbits.

Satellite constellation — coordinated group of satellites providing combined coverage or services.

Orbital period — time required for one complete orbit.

Orbital debris — human-made objects or fragments remaining in space that no longer serve a useful function.

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Key Takeaways

• Orbits can be classified by shape, altitude, inclination, period and orientation.
• Circular orbits have constant orbital radius and constant orbital speed.
• Elliptical orbits have changing radius and changing speed.
• The central body lies at one focus of an elliptical orbit.
• Satellites move fastest near periapsis and slowest near apoapsis.
• LEO satellites are relatively close to Earth and travel rapidly.
• LEO offers low communication delay and can provide high-resolution Earth observation.
• GEO satellites orbit approximately 35,786 km above Earth's surface.
• A geostationary satellite appears approximately stationary over one longitude.
• GEO is particularly useful for communications and continuous weather monitoring.
• Polar satellites can observe nearly the entire Earth as Earth rotates beneath their orbital paths.
• Sun-synchronous orbits provide similar lighting conditions on repeated observations.
• MEO is widely useful for navigation satellite constellations.
• Highly elliptical orbits can provide long periods of coverage over high latitudes.
• LEO and polar are not mutually exclusive; a satellite can be both.
• Elliptical transfer orbits allow spacecraft to move between different orbital altitudes.
• Different missions require different orbital characteristics.
• Low altitude improves some forms of imaging and reduces communication distance but reduces instantaneous coverage.
• High altitude increases coverage but generally increases signal delay and launch requirements.
• Orbit selection is an engineering trade-off involving coverage, resolution, latency, energy, lifetime, debris risk and mission objectives.
• The most appropriate orbit is the one that best supports the satellite's intended mission.

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Check Your Understanding

1. Explain the main difference between a circular orbit and an elliptical orbit.
2. Define periapsis and apoapsis.
3. Explain why a satellite in an elliptical orbit moves faster near periapsis than near apoapsis.
4. What does orbital eccentricity describe? What is the eccentricity of a circular orbit?
5. Compare LEO and GEO in terms of altitude, orbital speed, period, coverage and communication delay.
6. Explain why a geostationary satellite must orbit above Earth's equator.
7. Why are geostationary satellites particularly useful for communications and continuous weather monitoring?
8. Explain how a polar satellite can eventually observe almost the entire Earth even though it follows approximately the same orbital plane during each revolution.
9. Why are Sun-synchronous orbits useful for comparing images of Earth's surface taken on different days?
10. A mission needs high-resolution images of forests around the world under similar lighting conditions. Recommend an orbit and justify your choice.
11. A satellite must continuously monitor tropical storms over the same large region. Recommend an orbit and explain your reasoning.
12. Explain why navigation systems often use constellations of satellites rather than one satellite.
13. Describe how an elliptical transfer orbit can be used to move a spacecraft from a low circular orbit to a higher circular orbit.
14. Explain why a highly elliptical orbit might be preferable to GEO for communications with very high-latitude regions.
15. A company proposes putting every type of satellite into geostationary orbit because "GEO covers more of Earth, so it must always be better." Evaluate this proposal using at least four factors that engineers must consider when selecting an orbit.

Orbital Energy

Orbital energy describes the mechanical energy of an object moving in a gravitational field. For a satellite, planet, moon, or spacecraft, the two main forms of mechanical energy are:

• kinetic energy (Eₖ) because the object is moving
• gravitational potential energy (Eₚ) because of its position in the gravitational field

The total orbital energy is:

E_total = Eₖ + Eₚ

For an object of mass m orbiting a much larger central body of mass M:

Eₖ = 1/2 mv²

Eₚ = −GMm/r

Therefore:

E_total = 1/2 mv² − GMm/r

The sign and magnitude of this total energy tell us a great deal about the object's orbit.

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Kinetic Energy in Orbit

Any orbiting object has kinetic energy because it is moving.

Eₖ = 1/2 mv²

where:

• m = mass of the orbiting object
• v = orbital speed

Kinetic energy is always:

Eₖ ≥ 0

For an orbiting satellite with nonzero speed:

Eₖ > 0

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Worked Example: Satellite Kinetic Energy

A satellite has:

m = 1000 kg

v = 7500 m/s

Then:

Eₖ = 1/2(1000)(7500²)

Eₖ = 2.81 × 10¹⁰ J

Therefore, the satellite has approximately:

28.1 billion joules of kinetic energy.

Orbital speeds are extremely high, so even relatively small satellites can have enormous kinetic energies.

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Gravitational Potential Energy

For two masses separated by distance r:

Eₚ = −GMm/r

where:

• G = 6.67 × 10⁻¹¹ N·m²/kg²
• M = mass of the central body
• m = mass of the orbiting object
• r = distance between their centres

Unlike the near-surface equation:

ΔEₚ = mgΔh

the equation:

Eₚ = −GMm/r

is appropriate when gravitational field strength changes significantly with distance.

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Why Gravitational Potential Energy Is Negative

For orbital calculations, gravitational potential energy is conventionally defined as:

Eₚ = 0 at infinite separation

A satellite at any finite distance from a planet is gravitationally bound to it.

Energy must be supplied to move the satellite from that position all the way to infinity.

Therefore:

Eₚ < 0

The negative sign represents the fact that the two objects form a gravitationally bound system relative to the zero-at-infinity reference.

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Potential Energy Becomes Less Negative with Distance

Consider:

Eₚ = −GMm/r

As r increases, the magnitude of Eₚ decreases.

For example:

Moving farther away increases gravitational potential energy.

It becomes:

less negative

and approaches zero.

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Total Orbital Energy

Total mechanical energy is:

E_total = Eₖ + Eₚ

or:

E_total = 1/2 mv² − GMm/r

The value of E_total can be:

• negative
• zero
• positive

These three possibilities correspond to fundamentally different gravitational states.

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Negative Total Energy

If:

E_total < 0

the object is gravitationally bound.

It does not have enough mechanical energy to escape to infinity.

Bound gravitational orbits include:

• circular orbits
• elliptical orbits

Therefore:

bound orbit → negative total orbital energy

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Zero Total Energy

If:

E_total = 0

the object is exactly at the ideal escape threshold.

It can move infinitely far from the central body while its speed approaches zero.

This corresponds to the minimum escape condition.

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Positive Total Energy

If:

E_total > 0

the object is unbound.

It can escape to infinity while retaining kinetic energy.

Therefore:

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Energy in a Circular Orbit

Circular orbits have an especially useful relationship between kinetic, potential, and total energy.

For circular motion:

F_gravity = F_centripetal

Therefore:

GMm/r² = mv²/r

Cancel m and simplify:

v² = GM/r

Now substitute this into kinetic energy:

Eₖ = 1/2 mv²

Eₖ = 1/2 m(GM/r)

Therefore:

Eₖ = GMm/(2r)

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Potential Energy in a Circular Orbit

Gravitational potential energy is:

Eₚ = −GMm/r

Compare this with kinetic energy:

Eₖ = GMm/(2r)

Therefore:

Eₚ = −2Eₖ

This is an important circular-orbit relationship.

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Total Energy in a Circular Orbit

Total energy is:

E_total = Eₖ + Eₚ

Substitute:

E_total = GMm/(2r) − GMm/r

Therefore:

E_total = −GMm/(2r)

So for a circular orbit:

Eₖ = +GMm/(2r)

Eₚ = −GMm/r

E_total = −GMm/(2r)

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The 1 : −2 : −1 Relationship

For a circular orbit, the energies have the ratio:

Eₖ : Eₚ : E_total = 1 : −2 : −1

For example, if:

Eₖ = +4 × 10⁹ J

then:

Eₚ = −8 × 10⁹ J

and:

E_total = −4 × 10⁹ J

This provides a useful shortcut for circular-orbit calculations.

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Worked Circular-Orbit Example

A 500 kg satellite orbits Earth at:

r = 7.00 × 10⁶ m

Use:

G = 6.67 × 10⁻¹¹ N·m²/kg²

M_Earth = 5.97 × 10²⁴ kg

First calculate:

GM ≈ 3.98 × 10¹⁴ m³/s²

Kinetic Energy

Eₖ = GMm/(2r)

Eₖ = (3.98 × 10¹⁴)(500) / [2(7.00 × 10⁶)]

Eₖ ≈ 1.42 × 10¹⁰ J

Potential Energy

Eₚ = −GMm/r

Eₚ ≈ −2.84 × 10¹⁰ J

Total Energy

E_total = −GMm/(2r)

E_total ≈ −1.42 × 10¹⁰ J

Check:

Eₖ + Eₚ

= 1.42 × 10¹⁰ − 2.84 × 10¹⁰

= −1.42 × 10¹⁰ J

The negative total energy confirms that the satellite is gravitationally bound.

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Why Bound Orbits Have Negative Energy

Imagine a satellite in orbit around Earth.

To remove it completely from Earth's gravitational influence, energy must be supplied.

At infinity:

Eₚ → 0

For the minimum escape condition:

Eₖ → 0

Therefore:

E_total → 0

If the satellite's current total energy is negative, additional energy must be supplied to raise its energy to zero.

This is why negative energy indicates a bound gravitational system.

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How Strongly Is an Object Bound?

The magnitude of the negative total energy indicates how strongly the object is bound.

For example:

E_total = −10¹¹ J

represents a more tightly bound orbit than:

E_total = −10⁹ J

assuming the same orbiting mass and central body.

The more negative the energy:

the more energy must be added to escape.

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Orbital Energy and Radius

For a circular orbit:

E_total = −GMm/(2r)

Therefore:

E_total ∝ −1/r

As orbital radius increases, total energy becomes:

less negative

and approaches zero.

So:

higher orbit → greater total energy

This is true even though the satellite's orbital speed becomes lower.

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Comparing Circular Orbits

Suppose Satellite A orbits at radius r.

Its total energy is:

E_A = −GMm/(2r)

Satellite B orbits at:

2r

Then:

E_B = −GMm/(4r)

Therefore:

E_B = 1/2 E_A

in magnitude.

If:

E_A = −100 MJ

then:

E_B = −50 MJ

The higher orbit has greater total energy because −50 MJ is greater than −100 MJ.

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Higher Orbit but Lower Speed

This can initially seem contradictory.

A satellite in a higher circular orbit has:

• lower orbital speed
• lower kinetic energy
• greater gravitational potential energy
• greater total mechanical energy

The increase in gravitational potential energy is larger than the decrease in kinetic energy.

Therefore, energy must be added to move a satellite into a higher orbit.

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Energy Changes Between Circular Orbits

Suppose a satellite moves from:

r₁ → r₂

where:

r₂ > r₁

Initial total energy:

E₁ = −GMm/(2r₁)

Final total energy:

E₂ = −GMm/(2r₂)

The required change is:

ΔE = E₂ − E₁

Because the higher orbit has less-negative energy:

ΔE > 0

Energy must be supplied.

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Worked Example: Raising an Orbit

Suppose a satellite has:

E₁ = −8.0 × 10⁹ J

in its original circular orbit.

Its new circular orbit has:

E₂ = −5.0 × 10⁹ J

Then:

ΔE = E₂ − E₁

= −5.0 × 10⁹ − (−8.0 × 10⁹)

= +3.0 × 10⁹ J

The spacecraft must gain:

3.0 × 10⁹ J

of orbital energy.

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Lowering an Orbit

Suppose instead:

E₁ = −5.0 × 10⁹ J

E₂ = −8.0 × 10⁹ J

Then:

ΔE = −8.0 × 10⁹ − (−5.0 × 10⁹)

= −3.0 × 10⁹ J

The satellite must lose:

3.0 × 10⁹ J

of orbital energy.

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Energy in Elliptical Orbits

In an elliptical orbit:

• orbital radius changes
• speed changes
• kinetic energy changes
• gravitational potential energy changes

However, if gravity is the only force doing work:

total orbital energy remains constant.

---

At Periapsis

At the closest point:

• r is smallest
• Eₚ is most negative
• speed is greatest
• Eₖ is greatest

---

At Apoapsis

At the farthest point:

• r is largest
• Eₚ is less negative
• speed is smallest
• Eₖ is smallest

Yet:

E_total remains constant

throughout the orbit.

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Energy Transformation in an Elliptical Orbit

As the satellite moves from apoapsis toward periapsis:

gravitational potential energy → kinetic energy

As it moves from periapsis toward apoapsis:

kinetic energy → gravitational potential energy

This is similar to an object falling and rising near Earth's surface, but the gravitational field changes substantially throughout the orbit.

---

Total Energy of an Elliptical Orbit

For an elliptical orbit:

E_total = −GMm/(2a)

where:

a = semi-major axis

The semi-major axis describes the overall size of the ellipse.

This equation is extremely important because total orbital energy depends on the semi-major axis, not the satellite's instantaneous distance from the central body.

---

Circular Orbit as a Special Case

For a circular orbit:

a = r

Therefore:

E_total = −GMm/(2r)

So the circular-orbit equation is simply a special case of the more general elliptical-orbit equation.

---

Same Semi-Major Axis, Same Total Energy

Two elliptical orbits with different shapes can have the same semi-major axis.

If they orbit the same central body and involve the same satellite mass, they have the same total orbital energy.

Their kinetic and potential energies at particular locations may differ, but their total orbital energy is identical.

---

Specific Orbital Energy

It is often useful to calculate energy per kilogram of spacecraft.

Specific orbital energy is:

ε = E_total/m

For an object at distance r with speed v:

ε = v²/2 − GM/r

For a circular orbit:

ε = −GM/(2r)

For an elliptical orbit:

ε = −GM/(2a)

The unit is:

J/kg

which is equivalent to:

m²/s²

Using specific energy removes the spacecraft mass from the calculation.

---

The Vis-Viva Equation

Orbital energy can also be used to determine speed anywhere in an elliptical orbit.

Start with:

E_total = 1/2 mv² − GMm/r

For an elliptical orbit:

E_total = −GMm/(2a)

Set them equal:

1/2 mv² − GMm/r = −GMm/(2a)

Cancel m and rearrange:

v² = GM(2/r − 1/a)

Therefore:

v = √[GM(2/r − 1/a)]

This is called the vis-viva equation.

---

What Vis-Viva Tells Us

For a particular orbit, a is constant.

As r decreases:

v increases

As r increases:

v decreases

This mathematically explains why objects move:

• fastest at periapsis
• slowest at apoapsis

For a circular orbit:

a = r

so:

v² = GM(2/r − 1/r)

= GM/r

giving:

v = √(GM/r)

as expected.

---

Orbital Transfers

A spacecraft changes orbit by changing its orbital energy.

Rocket engines do this by changing the spacecraft's velocity.

Since:

Eₖ = 1/2 mv²

an engine burn can alter kinetic energy and therefore total orbital energy.

A spacecraft can:

• add orbital energy to move toward a higher orbit
• remove orbital energy to move toward a lower orbit

---

Raising an Orbit

Suppose a spacecraft is travelling in a circular orbit.

It fires its engine forward, in the direction of travel.

Immediately:

• speed increases
• kinetic energy increases
• total orbital energy increases

The spacecraft does not instantly jump to a higher circular orbit.

Instead, it enters an elliptical transfer orbit.

Its original location becomes the low point of the new ellipse.

---

Climbing Through the Transfer Orbit

After the engine shuts off, the spacecraft moves outward.

As it climbs:

• r increases
• gravitational potential energy increases
• kinetic energy decreases
• speed decreases

Total energy remains approximately constant during the unpowered coast.

The spacecraft eventually reaches the high point of the transfer orbit.

---

Circularizing the Higher Orbit

At the high point, another forward engine burn increases the spacecraft's speed.

This raises the opposite side of the orbit.

If the burn is correct, the elliptical trajectory becomes a new higher circular orbit.

This produces an interesting result:

energy was added, yet the final orbital speed is lower than in the original low orbit.

---

Why This Is Not a Contradiction

The final higher orbit has:

• lower kinetic energy
• greater gravitational potential energy
• greater total energy

The increase in potential energy is larger than the decrease in kinetic energy.

Therefore:

higher orbit = greater total orbital energy

even though:

higher circular orbit = lower orbital speed

---

Hohmann Transfer

A Hohmann transfer is an efficient way of moving between two circular orbits in the same plane.

It uses an elliptical transfer orbit.

A simplified transfer to a higher orbit involves:

1. initial circular orbit
2. forward engine burn
3. elliptical transfer orbit
4. coast toward apoapsis
5. second forward burn
6. final higher circular orbit

Both burns add orbital energy.

---

Energy During a Hohmann Transfer

Suppose:

r₁ < r₂

The initial circular orbit has:

E₁ = −GMm/(2r₁)

The transfer ellipse has semi-major axis:

a = (r₁ + r₂)/2

Therefore:

E_transfer = −GMm/(2a)

The final circular orbit has:

E₂ = −GMm/(2r₂)

The energy ordering is:

E₁ < E_transfer < E₂

Remember that these energies are negative.

For example:

−100 MJ < −70 MJ < −50 MJ

The spacecraft becomes progressively less tightly bound.

---

Energy Diagram for Orbit Raising

A useful conceptual sequence is:

low circular orbit

E_total = very negative

↓

engine adds energy

↓

transfer ellipse

E_total = less negative

↓

engine adds more energy

↓

high circular orbit

E_total = even less negative

The spacecraft is still bound as long as:

E_total < 0

---

Lowering an Orbit

To lower an orbit, the spacecraft can fire its engine opposite its direction of motion.

Immediately:

• speed decreases
• kinetic energy decreases
• total orbital energy decreases
• energy becomes more negative

The spacecraft enters an ellipse with a lower point on the opposite side.

A second burn can then circularize the lower orbit.

---

Deorbiting

To deliberately return a spacecraft to Earth, engineers can perform a deorbit burn.

The spacecraft fires its engine to reduce orbital energy.

Its new orbit intersects Earth's atmosphere.

Atmospheric drag then removes additional mechanical energy.

The spacecraft descends and re-enters.

---

Atmospheric Drag and Orbital Energy

Satellites in low Earth orbit encounter traces of atmosphere.

Drag does negative work on the satellite.

Therefore:

orbital energy decreases

and becomes more negative.

The satellite gradually moves into a lower orbit.

This process is called:

orbital decay

---

Why a Decaying Satellite Can Speed Up

This seems strange.

Drag slows the satellite, yet lower circular orbits have higher orbital speeds.

The sequence is important.

Drag first removes mechanical energy.

The orbit becomes elliptical and drops lower.

As the satellite falls:

• gravitational potential energy decreases
• some potential energy becomes kinetic energy

The satellite can eventually move faster in the lower orbit even though its total orbital energy has decreased.

This is an important example of why speed alone does not determine orbital energy.

---

Escape as an Orbital-Energy Problem

A bound circular orbit has:

E_total < 0

To escape:

E_total must reach at least 0

Therefore, engines must add enough energy to move the spacecraft from negative orbital energy to zero or positive energy.

At the exact escape threshold:

E_total = 0

---

Energy Needed to Escape from a Circular Orbit

For a circular orbit:

E_initial = −GMm/(2r)

At the minimum escape condition:

E_final = 0

Therefore:

ΔE = E_final − E_initial

ΔE = 0 − [−GMm/(2r)]

So:

ΔE = GMm/(2r)

Interestingly, this equals the satellite's existing circular-orbit kinetic energy.

---

Orbital Speed and Escape Speed

For a circular orbit:

v_orbit = √(GM/r)

Escape speed at the same radius is:

v_escape = √(2GM/r)

Therefore:

v_escape = √2 v_orbit

An orbiting spacecraft already possesses substantial kinetic energy, so it does not need to acquire the full escape energy from rest at that location.

---

Worked Escape Example

Suppose a spacecraft has circular orbital speed:

7.5 km/s

Then:

v_escape = √2(7.5)

≈ 10.6 km/s

Increasing its speed in the appropriate direction can raise its total energy from negative toward zero.

At:

10.6 km/s

at that same radius, the spacecraft would be at the ideal local escape threshold.

---

Bound and Unbound Trajectories

Orbital energy allows trajectories to be classified.

Circular Orbit

E_total < 0

Bound.

Elliptical Orbit

E_total < 0

Bound.

Escape Threshold

E_total = 0

Ideal limiting escape trajectory.

Hyperbolic Trajectory

E_total > 0

Unbound.

---

Orbital Energy and Gravity Assists

Gravity assists can also change a spacecraft's orbital energy relative to the Sun.

During a planetary flyby, the planet bends the spacecraft's velocity vector.

Because the planet itself is moving around the Sun, the spacecraft can:

• gain heliocentric orbital energy
• lose heliocentric orbital energy

This can move the spacecraft onto a different solar orbit without requiring the equivalent energy change entirely from onboard propulsion.

Thus orbital energy provides another way of understanding gravitational slingshots.

---

Real Mission Applications

Orbital-energy calculations are important for:

• satellite deployment
• orbit raising
• geostationary transfers
• lunar missions
• interplanetary missions
• planetary orbit insertion
• rendezvous
• deorbiting
• gravity assists
• asteroid and comet missions

Mission planners continually analyze how much orbital energy must be added or removed.

---

Orbit Insertion

A spacecraft arriving at another planet may initially have:

E_total > 0

relative to that planet.

It is therefore unbound and would fly past.

To enter orbit, the spacecraft must lose energy.

An engine burn can reduce kinetic energy until:

E_total < 0

The spacecraft then becomes gravitationally bound.

This manoeuvre is called:

orbit insertion.

---

Aerobraking and Energy

If a planet has an atmosphere, spacecraft may sometimes use atmospheric drag to remove orbital energy.

This is called aerobraking.

Instead of using large amounts of propellant, repeated atmospheric passes can gradually lower the spacecraft's orbital energy.

The lost mechanical energy is transferred mainly into thermal energy.

---

Comparing Two Orbits

Suppose two identical satellites orbit the same planet.

Satellite A:

r = R

Satellite B:

r = 2R

For circular orbits:

Kinetic Energy

Eₖ ∝ 1/r

Therefore:

Eₖ,B = 1/2 Eₖ,A

Potential Energy

Eₚ ∝ −1/r

Therefore, Satellite B's potential energy has half the negative magnitude.

Total Energy

E_total ∝ −1/r

Therefore, Satellite B's total energy also has half the negative magnitude.

Satellite B is less tightly bound.

---

Energy Graphs

Orbital energy can be represented graphically as a function of distance.

Gravitational potential energy:

Eₚ = −GMm/r

approaches:

0

as:

r → ∞

For a circular orbit:

Eₖ = +GMm/(2r)

and:

E_total = −GMm/(2r)

At any circular-orbit radius:

• Eₚ is negative
• Eₖ is positive
• |Eₚ| = 2Eₖ
• E_total is negative

As radius increases, all three approach zero.

---

Reading an Orbital-Energy Diagram

Suppose an energy diagram shows:

Eₚ = −80 MJ

For a circular orbit:

Eₖ = +40 MJ

and:

E_total = −40 MJ

If energy is added until:

E_total = −10 MJ

the spacecraft remains bound.

If energy is added until:

E_total = 0

it reaches the escape threshold.

If:

E_total = +10 MJ

the spacecraft is unbound.

---

Orbital Energy Is Relative to the Central Body

Orbital energy depends on the gravitational system being considered.

A spacecraft may be:

• unbound from Earth
• but still bound to the Sun

For example, a spacecraft that escapes Earth usually enters a new orbit around the Sun.

Therefore, statements such as "the spacecraft has escaped" must specify:

escaped from what?

---

Leaving Earth Versus Leaving the Solar System

Escaping Earth requires enough energy to become unbound from Earth.

Escaping the Solar System requires enough heliocentric energy to become unbound from the Sun.

These are different energy conditions.

A spacecraft can have:

E_Earth ≥ 0

while still having:

E_Sun < 0

---

Energy and Mission Planning

When planning an orbital manoeuvre, engineers can ask:

What is the initial orbital energy?

What is the desired final orbital energy?

How much energy must be added or removed?

What velocity change will produce that energy change?

Where in the orbit should the engine burn occur?

Because kinetic energy depends on v², the location of a burn can strongly affect the resulting orbit.

---

Why Burn Location Matters

A spacecraft's speed changes around an elliptical orbit.

An engine burn performed where the spacecraft is already moving rapidly can produce a particularly useful change in orbital energy.

This is one reason orbital manoeuvres are carefully timed rather than performed at arbitrary locations.

Mission design involves not only:

how much thrust

but also:

where and when the thrust is applied.

---

A General Problem-Solving Strategy

For an orbital-energy problem:

Identify the system

Determine:

• central mass M
• spacecraft mass m
• orbital radius r
• orbital speed v

Calculate kinetic energy if needed

Eₖ = 1/2 mv²

Calculate gravitational potential energy

Eₚ = −GMm/r

Calculate total energy

E_total = Eₖ + Eₚ

For circular orbits, use the shortcuts:

Eₖ = GMm/(2r)

Eₚ = −GMm/r

E_total = −GMm/(2r)

Interpret the sign

• negative → bound
• zero → escape threshold
• positive → unbound

For an orbital transfer

Calculate:

ΔE = E_final − E_initial

Then determine whether energy must be:

• added
• removed

---

Common Mistakes

Forgetting that gravitational potential energy is negative

For orbital problems:

Eₚ = −GMm/r

Using mgh for large orbital changes

Use −GMm/r when gravitational field strength changes significantly.

Thinking negative energy means the satellite has no energy

Negative total energy means the satellite is gravitationally bound relative to the chosen zero.

Thinking higher orbits have less total energy

Higher circular orbits have greater, less-negative total energy.

Thinking higher orbital energy means higher circular speed

Higher circular orbits have greater total energy but lower orbital speed.

Confusing kinetic energy with total energy

Total orbital energy includes both kinetic and gravitational potential energy.

Thinking an elliptical orbit has changing total energy

Eₖ and Eₚ change, but their sum remains constant when only gravity does work.

Using r as altitude

Orbital radius is measured from the centre of the central body.

Forgetting the factor of 2

For circular orbits:

E_total = −GMm/(2r)

not −GMm/r.

Thinking zero total energy means the spacecraft is stationary

At a finite distance, an object at the escape threshold is moving. Its speed approaches zero only as r approaches infinity.

---

Key Terms

Orbital energy — mechanical energy associated with an object's motion and position in a gravitational system.

Kinetic energy — energy due to motion, calculated using Eₖ = 1/2mv².

Gravitational potential energy — energy associated with the positions of masses in a gravitational field, Eₚ = −GMm/r.

Total orbital energy — sum of kinetic and gravitational potential energy.

Bound orbit — orbit with negative total mechanical energy.

Unbound trajectory — trajectory with positive total mechanical energy relative to the gravitational source.

Escape threshold — condition where total mechanical energy equals zero.

Specific orbital energy — orbital energy per unit spacecraft mass.

Semi-major axis — half the longest diameter of an ellipse; determines the total energy of a bound Keplerian orbit.

Periapsis — closest point of an orbit to the central body.

Apoapsis — farthest point of an orbit from the central body.

Vis-viva equation — equation relating orbital speed, distance and semi-major axis.

Orbital transfer — manoeuvre that changes a spacecraft from one orbit to another.

Hohmann transfer — efficient two-burn transfer between approximately circular coplanar orbits.

Orbit insertion — manoeuvre that reduces spacecraft energy sufficiently for gravitational capture.

Deorbit burn — manoeuvre that decreases orbital energy so a spacecraft's trajectory enters an atmosphere or intersects the central body.

Orbital decay — gradual loss of orbital energy, often due to atmospheric drag.

Aerobraking — use of atmospheric drag to remove spacecraft orbital energy.

---

Key Takeaways

• Orbital energy consists primarily of kinetic energy and gravitational potential energy.
• E_total = Eₖ + Eₚ.
• Kinetic energy is Eₖ = 1/2mv².
• Gravitational potential energy is Eₚ = −GMm/r.
• Gravitational potential energy is negative because zero is defined at infinite separation.
• Bound circular and elliptical orbits have negative total energy.
• Zero total energy represents the ideal escape threshold.
• Positive total energy represents an unbound trajectory.
• For a circular orbit, Eₖ = GMm/(2r).
• For a circular orbit, Eₚ = −GMm/r.
• For a circular orbit, E_total = −GMm/(2r).
• Therefore, for circular orbits, Eₚ = −2Eₖ.
• Higher circular orbits have lower speeds but greater total orbital energy.
• Moving to a higher orbit requires adding energy.
• Moving to a lower orbit requires removing energy.
• In an elliptical orbit, kinetic and potential energy continually transform into one another.
• Total mechanical energy remains constant during an unpowered ideal orbit.
• Objects move fastest near periapsis and slowest near apoapsis.
• For an elliptical orbit, E_total = −GMm/(2a).
• The vis-viva equation relates orbital speed to position within an elliptical orbit.
• Rocket burns change kinetic energy and therefore change orbital energy.
• Orbital transfers commonly use temporary elliptical trajectories.
• Reaching escape conditions means raising total orbital energy to zero or above.
• Gravity assists can change a spacecraft's orbital energy relative to the Sun.
• Orbital-energy analysis is fundamental to satellite operations, interplanetary travel, orbit insertion, aerobraking and spacecraft return.

---

Check Your Understanding

1. Define orbital energy and identify its two main components.
2. Explain why gravitational potential energy is negative for an orbiting satellite.
3. A 1200 kg satellite moves at 7.4 × 10³ m/s. Calculate its kinetic energy.
4. A satellite has Eₖ = +3.0 × 10¹⁰ J and Eₚ = −6.0 × 10¹⁰ J. Calculate its total energy and determine whether it is gravitationally bound.
5. Derive Eₖ = GMm/(2r) for a circular orbit using the circular orbital-speed relationship.
6. A circular-orbit satellite has kinetic energy +8.0 × 10⁹ J. Determine its gravitational potential energy and total orbital energy.
7. Explain why a bound orbit must have negative total mechanical energy.
8. Two identical satellites orbit the same planet at radii r and 2r. Compare their kinetic, potential and total energies.
9. Explain why a satellite in a higher circular orbit can have less kinetic energy but more total energy than a satellite in a lower orbit.
10. A spacecraft moves from an orbit with total energy −9.0 × 10⁹ J to one with total energy −5.0 × 10⁹ J. Calculate the change in orbital energy and state whether energy was added or removed.
11. Describe the changes in kinetic and gravitational potential energy as a satellite moves from apoapsis to periapsis in an elliptical orbit.
12. Explain the significance of the semi-major axis in the equation E_total = −GMm/(2a).
13. Describe the two main engine burns used to move a spacecraft from a lower circular orbit to a higher circular orbit using a Hohmann transfer.
14. A spacecraft approaching a planet has positive total orbital energy relative to that planet. Explain what must happen for the spacecraft to enter a bound orbit.
15. A student says, "A spacecraft in a higher circular orbit has less speed, so it must have less total energy." Evaluate this statement using kinetic energy, gravitational potential energy and total orbital energy.

Kepler’s Laws of Planetary Motion

In the early 1600s, Johannes Kepler developed three mathematical laws describing how planets move around the Sun. His laws were based largely on the extremely precise astronomical observations collected by Tycho Brahe.

Kepler's laws showed that planetary motion follows predictable mathematical patterns:

• planets travel in elliptical orbits
• planets change speed as they move around the Sun
• orbital period is related to orbital size

Later, Newton showed that Kepler's laws can be explained using gravity and the laws of motion.

---

Kepler's Three Laws

Kepler's laws can be summarized as:

First Law — Law of Ellipses

Planets move in elliptical orbits with the Sun at one focus.

Second Law — Law of Equal Areas

A line joining a planet and the Sun sweeps out equal areas during equal time intervals.

Third Law — Law of Periods

The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

These three laws describe the shape, changing speed, and timing of planetary orbits.

---

Kepler's First Law — The Law of Ellipses

Kepler's First Law states:

A planet moves in an elliptical orbit with the Sun at one focus of the ellipse.

Before Kepler, circular motion was commonly assumed to be the natural shape of planetary orbits.

Kepler's analysis showed that an ellipse provides a much better description.

---

Understanding an Ellipse

An ellipse is a closed curve with two special points called foci.

The Sun is located at:

one focus

It is not generally at the geometric centre of the ellipse.

The other focus contains no physical object simply because it is a focus.

---

Semi-Major Axis

The longest diameter of an ellipse is called the major axis.

Half of this distance is the:

semi-major axis, a

The semi-major axis provides a useful measure of the overall size of an orbit.

It is especially important in Kepler's Third Law.

---

Perihelion and Aphelion

For an object orbiting the Sun:

perihelion = closest point to the Sun

aphelion = farthest point from the Sun

At perihelion:

r is smallest

At aphelion:

r is largest

Because gravitational force depends on distance, the strength of the Sun's gravitational pull changes throughout an elliptical orbit.

---

Eccentricity

The shape of an ellipse can be described using eccentricity, e.

For a circle:

e = 0

For an ellipse:

0 < e < 1

As e approaches 1, the ellipse becomes increasingly stretched.

Most planetary orbits in our Solar System have relatively small eccentricities and therefore look fairly close to circular.

---

Circular Orbits and Kepler's First Law

A circle is a special case of an ellipse.

Therefore, an approximately circular planetary orbit is still consistent with Kepler's First Law.

For a perfectly circular orbit:

e = 0

and the two foci coincide at the centre.

---

Kepler's Second Law — Equal Areas

Kepler's Second Law states:

A line joining a planet to the Sun sweeps out equal areas in equal time intervals.

This law describes how a planet's orbital speed changes.

---

What Does "Sweeps Out Area" Mean?

Imagine drawing a line from the Sun to a planet.

As the planet moves, the line moves with it.

During a certain amount of time, such as 30 days, the moving line sweeps across a region.

Kepler discovered that equal time intervals produce:

equal swept areas

even though the planet travels different distances along its orbit.

---

Planetary Speed Changes

Near perihelion, the planet is close to the Sun.

It must travel a relatively large distance along its orbit during a given time interval to sweep out the required area.

Therefore:

planet moves faster near perihelion

Near aphelion, the planet is farther from the Sun.

It travels a shorter distance during the same time interval.

Therefore:

planet moves slower near aphelion

---

Perihelion Versus Aphelion

So:

closer to Sun → faster

farther from Sun → slower

This is consistent with the energy changes in an elliptical orbit.

---

Energy and Kepler's Second Law

As a planet moves toward the Sun:

• gravitational potential energy decreases
• kinetic energy increases
• orbital speed increases

As the planet moves away:

• kinetic energy decreases
• gravitational potential energy increases
• orbital speed decreases

Total mechanical energy remains constant in the ideal two-body model.

Therefore, Kepler's Second Law agrees with conservation of mechanical energy.

---

Equal Areas Do Not Mean Equal Distances

Suppose a planet spends 30 days near perihelion and another 30 days near aphelion.

The areas swept out are equal.

However, the distances travelled are not.

Near perihelion:

greater distance travelled

Near aphelion:

smaller distance travelled

This means the average speeds during those intervals are different.

---

Kepler's Second Law and Angular Momentum

Kepler's Second Law is closely connected to conservation of angular momentum.

For orbital motion, angular momentum can be written conceptually as:

L = mrv_t

where v_t is the tangential component of velocity.

For an ideal gravitational orbit:

angular momentum remains constant.

Gravity points toward the Sun, so it produces no torque about the Sun in the ideal two-body model.

Therefore:

r × tangential speed remains related in such a way that equal areas are swept in equal times.

---

A Useful Perihelion-Aphelion Relationship

At perihelion and aphelion, the velocity is perpendicular to the radial line.

Conservation of angular momentum gives:

r_p v_p = r_a v_a

where:

• r_p = perihelion distance
• v_p = perihelion speed
• r_a = aphelion distance
• v_a = aphelion speed

This relationship allows speeds at different parts of an elliptical orbit to be compared.

---

Worked Example: Perihelion and Aphelion Speed

Suppose a planet has:

r_p = 1.0 × 10¹¹ m

v_p = 40 km/s

r_a = 2.0 × 10¹¹ m

Use:

r_p v_p = r_a v_a

Therefore:

v_a = r_p v_p / r_a

v_a = (1.0 × 10¹¹)(40)/(2.0 × 10¹¹)

v_a = 20 km/s

The planet travels twice as fast at perihelion as at aphelion in this simplified example.

---

Kepler's Third Law — The Law of Periods

Kepler's Third Law states:

The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

Mathematically:

T² ∝ a³

where:

• T = orbital period
• a = semi-major axis

This law relates the time required to complete an orbit to the size of the orbit.

---

Meaning of the Third Law

Planets farther from the Sun take much longer to complete their orbits.

There are two important reasons:

• their orbital paths are larger
• their orbital speeds are generally lower

Therefore:

larger orbit → longer orbital period

---

Comparing Two Orbits

For objects orbiting the same central body:

T₁²/a₁³ = T₂²/a₂³

This allows two orbital periods or orbital sizes to be compared without needing the mass of the central body.

---

A Convenient Solar-System Form

For planets orbiting the Sun, if:

• T is measured in years
• a is measured in astronomical units (AU)

then approximately:

T² = a³

because Earth's orbit provides the reference:

Earth:

T = 1 year

a = 1 AU

Therefore:

1² = 1³

---

Astronomical Unit

One astronomical unit is approximately the average Earth-Sun distance:

1 AU ≈ 1.50 × 10¹¹ m

Astronomical units are convenient for comparing planetary distances within the Solar System.

---

Worked Example: Planet at 4 AU

Suppose a planet has:

a = 4 AU

Using:

T² = a³

T² = 4³

T² = 64

T = √64

T = 8 years

Therefore, a planet with semi-major axis 4 AU would take approximately 8 Earth years to orbit the Sun.

---

Worked Example: Finding Orbital Size

Suppose a planet takes:

27 years

to orbit the Sun.

Use:

T² = a³

27² = a³

729 = a³

a = ∛729

a = 9 AU

Therefore, its semi-major axis would be approximately 9 AU.

---

A More Realistic Example

Suppose a planet has:

a = 5.2 AU

Then:

T² = 5.2³

T² ≈ 140.6

T ≈ √140.6

T ≈ 11.9 years

This is approximately the orbital period of Jupiter.

---

Planetary Data

Approximate Solar System values illustrate Kepler's Third Law:

As orbital size increases, orbital period increases strongly.

---

Testing Kepler's Third Law with Data

Consider Mars:

a = 1.52 AU

T = 1.88 years

Calculate:

a³ = 1.52³

≈ 3.51

Now:

T² = 1.88²

≈ 3.53

These values are extremely close.

Therefore:

T² ≈ a³

as expected.

---

Interpreting a Kepler's Third Law Graph

If we plot:

T² against a³

for planets orbiting the same star, the data should form approximately a straight line through the origin.

This provides a useful way to test Kepler's Third Law using observational data.

---

What Would T Versus a Look Like?

A graph of T directly against a would not be a straight line.

Since:

T² ∝ a³

then:

T ∝ a^(3/2)

Therefore, T increases non-linearly with a.

This is why transforming the variables to T² and a³ is useful.

---

Connecting Kepler to Newton

Kepler's laws were originally empirical.

This means they were discovered from patterns in observational data.

Kepler could describe planetary motion accurately, but he did not have Newton's later theory of universal gravitation to fully explain why the planets behaved this way.

Newton later showed that Kepler's laws naturally follow from:

• Newton's laws of motion
• universal gravitation

---

Newton's Law of Universal Gravitation

Newton proposed:

F = GMm/r²

This gravitational force pulls a planet toward the Sun.

For a circular orbit, gravity supplies the centripetal force:

GMm/r² = mv²/r

Cancel m:

GM/r² = v²/r

Therefore:

v² = GM/r

and:

v = √(GM/r)

This explains why orbital speed decreases as orbital radius increases.

---

Deriving Kepler's Third Law with Newtonian Gravity

For a circular orbit:

v = 2πr/T

where T is the orbital period.

Gravity provides centripetal force:

GMm/r² = mv²/r

Substitute:

v = 2πr/T

Then:

GM/r² = (4π²r²/T²)/r

So:

GM/r² = 4π²r/T²

Rearrange:

T² = 4π²r³/(GM)

Therefore:

T² = (4π²/GM)r³

For objects orbiting the same central body, G and M are constant.

Therefore:

T² ∝ r³

For elliptical orbits, the corresponding result uses the semi-major axis:

T² = 4π²a³/(GM)

This is Newton's gravitational explanation of Kepler's Third Law.

---

Why the Orbiting Mass Cancels

Notice that the planet or satellite mass m cancels from the derivation.

Therefore, in the approximation where the central body's mass is much larger:

orbital period does not depend on the mass of the orbiting object.

A small satellite and a much larger satellite at the same orbital radius around Earth have the same ideal orbital period.

---

A More General Third-Law Equation

When the two orbiting masses are both important, the more general equation is:

T² = 4π²a³/[G(M + m)]

For a planet orbiting the Sun:

M_Sun ≫ m_planet

so:

M + m ≈ M

and the simpler equation works extremely well.

For binary stars with comparable masses, however, both masses should be included.

---

Using Kepler's Third Law for Satellites

Kepler's Third Law applies not only to planets.

It can describe:

• moons orbiting planets
• artificial satellites orbiting Earth
• planets orbiting stars
• binary stars
• some asteroid systems

For satellites orbiting Earth:

T² = 4π²r³/(GM_Earth)

for a circular orbit.

---

Worked Satellite Example

Suppose a satellite has orbital radius:

r = 7.00 × 10⁶ m

Use:

GM_Earth ≈ 3.986 × 10¹⁴ m³/s²

Then:

T² = 4π²(7.00 × 10⁶)³/(3.986 × 10¹⁴)

T ≈ 5.83 × 10³ s

Convert to minutes:

T ≈ 5830/60

≈ 97 minutes

This is typical of a low-Earth-orbit satellite.

---

Geostationary Orbit and Kepler's Third Law

A geostationary satellite must have an orbital period matching Earth's rotation.

Its period is approximately one sidereal day.

Kepler's Third Law determines the orbital radius required for this period.

The resulting orbital radius is approximately:

42,164 km from Earth's centre

Since Earth's radius is about 6378 km:

altitude ≈ 42,164 − 6378

≈ 35,786 km

This demonstrates a direct engineering application of Kepler's laws.

---

Kepler's First Law and Newtonian Gravity

Newtonian gravity also explains why bound orbits are ellipses.

An inverse-square central force:

F ∝ 1/r²

produces trajectories that belong to a family of curves called conic sections.

Depending on the object's total energy, possible trajectories include:

• circle
• ellipse
• parabola
• hyperbola

Bound planetary orbits are ellipses.

---

Energy and Orbit Shape

Orbital energy provides another connection:

E_total < 0 → bound ellipse

A circle is a special ellipse.

E_total = 0 → escape threshold

E_total > 0 → unbound trajectory

Thus Kepler's First Law connects naturally with gravitational energy.

---

Kepler's Second Law and Newtonian Gravity

Gravity always points approximately toward the centre of the Sun.

This makes gravity a central force.

Because the force acts along the radial line, its torque about the Sun is zero:

τ = 0

Therefore:

angular momentum is conserved.

Conservation of angular momentum produces Kepler's Second Law.

So:

Kepler's equal-area law is a consequence of conservation of angular momentum.

---

First Law, Second Law and Third Law Together

The three laws describe different aspects of orbital motion.

Together they provide a powerful mathematical description of planetary motion.

---

Interpreting Planetary Data

Suppose observations give:

Check:

Planet B:

T² = 8² = 64

a³ = 4³ = 64

Planet C:

T² = 27² = 729

a³ = 9³ = 729

The data follow:

T² = a³

and are therefore consistent with Kepler's Third Law for planets orbiting the same star.

---

Predicting an Unknown Planet's Period

Suppose astronomers discover a planet orbiting the Sun with:

a = 16 AU

Use:

T² = a³

T² = 16³

= 4096

T = √4096

= 64 years

The planet would take approximately 64 Earth years to complete one orbit.

---

Predicting an Unknown Orbital Radius

Suppose an object has:

T = 125 years

Then:

T² = a³

125² = a³

15,625 = a³

a = 25 AU

So the object's semi-major axis is approximately:

25 AU

---

Comparing Orbital Periods Without Solving Separately

Suppose:

a₂ = 4a₁

Since:

T² ∝ a³

then:

T₂²/T₁² = 4³

= 64

Therefore:

T₂/T₁ = 8

So:

T₂ = 8T₁

Increasing orbital size by a factor of 4 increases period by a factor of 8.

---

Useful Scaling Patterns

Since:

T ∝ a^(3/2)

we obtain:

This relationship is much stronger than a simple direct proportion.

---

Determining the Mass of a Star

Newton's version of Kepler's Third Law can be rearranged:

T² = 4π²a³/(GM)

Therefore:

M = 4π²a³/(GT²)

If astronomers measure:

• orbital period
• orbital size

they can calculate the mass of the central star.

This is one of the most powerful applications of orbital mechanics.

---

Measuring Invisible Mass

We do not need to place a star on a scale.

Instead, we observe how objects move around it.

Orbital motion reveals gravitational mass.

The same principle can be used to estimate masses of:

• stars
• planets
• galaxies
• black holes
• binary systems

---

Discovering Exoplanets

Kepler's laws are also important in the study of exoplanets—planets orbiting stars other than the Sun.

Astronomers can measure:

• orbital period
• changes in stellar brightness
• motion of the host star

Orbital models can then reveal information about the planet's:

• orbital size
• orbital shape
• speed
• possible mass

Kepler's laws therefore remain central to modern astronomy.

---

Kepler's Laws and Space Missions

Spacecraft also follow Keplerian orbits when they coast under gravity.

Mission planners use these principles when designing:

• Earth satellites
• transfer orbits
• lunar missions
• interplanetary trajectories
• planetary flybys
• gravity assists

A spacecraft travelling from Earth toward Mars, for example, can follow part of an elliptical orbit around the Sun.

---

Hohmann Transfers and Kepler

A Hohmann transfer uses an elliptical orbit connecting two approximately circular orbits.

The spacecraft:

1. leaves one circular orbit
2. enters an ellipse
3. coasts along the ellipse
4. reaches the second orbit
5. performs another engine burn

Kepler's laws help predict:

• trajectory shape
• changing spacecraft speed
• transfer time

---

Kepler's Laws and Gravity Assists

During a planetary flyby, a spacecraft temporarily follows a gravitationally curved trajectory around the planet.

Orbital mechanics based on:

• Newtonian gravity
• conservation of energy
• conservation of angular momentum
• Keplerian trajectories

helps engineers predict the spacecraft's path.

Thus Kepler's ideas continue to play a role in modern gravity-assist missions.

---

Limitations of the Simple Kepler Model

Kepler's laws work extremely well for ideal two-body systems.

Real Solar System motion is more complicated.

Planets can be affected by:

• other planets
• moons
• non-spherical mass distributions
• relativistic effects
• atmospheric drag for low satellites
• radiation pressure for spacecraft

These effects cause small deviations from simple Keplerian orbits.

---

Orbital Perturbations

A perturbation is a small deviation from an ideal orbit caused by an additional influence.

For example, Earth's orbit is mainly controlled by the Sun but is also slightly affected by:

• Jupiter
• Venus
• the Moon
• other Solar System bodies

Spacecraft navigation models therefore go beyond the simple two-body equations when high precision is required.

---

Reading Orbital Data Systematically

When given planetary data, first identify what is being compared.

For orbital shape:

look for:

• eccentricity
• perihelion
• aphelion

For changing speed:

look for:

• position within the orbit
• distance from the Sun
• equal-area intervals

For orbital period:

look for:

• T
• a
• T²
• a³

Then choose the appropriate Kepler law.

---

A Kepler Problem-Solving Strategy

For First Law questions:

• identify the ellipse
• locate the central body at one focus
• identify periapsis and apoapsis
• consider eccentricity

For Second Law questions:

• compare equal time intervals
• compare swept areas
• remember that speed is greatest near periapsis
• use angular momentum when numerical analysis is required

For Third Law questions:

use:

T₁²/a₁³ = T₂²/a₂³

or, for Solar System values in years and AU:

T² = a³

For Newtonian calculations:

T² = 4π²a³/(GM)

---

Common Mistakes

Thinking planets orbit in perfect circles

Planetary orbits are ellipses, although many are close to circular.

Putting the Sun at the centre of an elliptical orbit

The Sun is at one focus.

Thinking the second focus contains another star

It is simply a geometric point.

Thinking planets travel at constant speed

Planets move faster near perihelion and slower near aphelion.

Thinking equal areas mean equal distances

Equal areas are swept during equal times, but orbital distances travelled can differ.

Using orbital radius instead of semi-major axis for a general ellipse

Kepler's Third Law uses the semi-major axis a.

Forgetting to square T or cube a

The relationship is:

T² ∝ a³

not T ∝ a.

Thinking distant planets take longer only because their paths are larger

They also generally move more slowly.

Thinking Kepler explained the gravitational force

Kepler described the orbital patterns. Newton later provided the gravitational explanation.

Ignoring the central body's mass

The simple comparison T²/a³ = constant applies to objects orbiting the same central body.

---

Key Terms

Kepler's First Law — planets move in elliptical orbits with the Sun at one focus.

Kepler's Second Law — a line from the Sun to a planet sweeps out equal areas in equal times.

Kepler's Third Law — orbital period squared is proportional to semi-major axis cubed.

Ellipse — closed curve with two foci.

Focus — one of two special points defining an ellipse.

Semi-major axis — half the longest diameter of an ellipse and a measure of orbital size.

Eccentricity — measure of how stretched an ellipse is.

Perihelion — point in a solar orbit closest to the Sun.

Aphelion — point in a solar orbit farthest from the Sun.

Orbital period — time required to complete one orbit.

Astronomical unit (AU) — approximately the average Earth-Sun distance, about 1.50 × 10¹¹ m.

Angular momentum — conserved quantity associated with rotational or orbital motion.

Central force — force directed toward or away from a central point.

Keplerian orbit — ideal orbit described by Kepler's laws.

Perturbation — small deviation from an ideal orbit caused by additional forces or bodies.

Exoplanet — planet orbiting a star other than the Sun.

---

Key Takeaways

• Kepler developed three laws describing planetary motion from astronomical observations.
• The First Law states that planetary orbits are ellipses with the Sun at one focus.
• A circle is a special ellipse with eccentricity zero.
• The semi-major axis measures the overall size of an orbit.
• The Second Law states that equal areas are swept out in equal times.
• Planets move fastest near perihelion.
• Planets move slowest near aphelion.
• Kepler's Second Law is connected to conservation of angular momentum.
• The Third Law states T² ∝ a³.
• More distant planets have much longer orbital periods.
• For Solar System calculations using years and AU, approximately T² = a³.
• For two objects orbiting the same body, T₁²/a₁³ = T₂²/a₂³.
• Newtonian gravity explains Kepler's laws.
• Newton's form of the Third Law is T² = 4π²a³/(GM) when the orbiting object's mass is negligible.
• Orbital period can be used to determine the mass of a star or planet.
• Kepler's laws apply to planets, moons, artificial satellites and many binary systems.
• They are used in modern spacecraft trajectory design.
• Real orbits can deviate slightly from ideal Keplerian motion because of perturbations.
• Kepler's laws transformed planetary motion from a geometric description into a quantitative mathematical science and helped provide the foundation for Newtonian gravitation.

---

Check Your Understanding

1. State Kepler's three laws of planetary motion.
2. According to Kepler's First Law, where is the Sun located within a planet's elliptical orbit?
3. Define semi-major axis and eccentricity.
4. Explain why a planet travels faster near perihelion than near aphelion.
5. A planet sweeps out an area of 3.0 × 10²¹ m² during 20 days near perihelion. What area should it sweep out during 20 days near aphelion?
6. A planet has perihelion distance 1.2 × 10¹¹ m and perihelion speed 40 km/s. Its aphelion distance is 2.0 × 10¹¹ m. Using conservation of angular momentum, calculate its aphelion speed.
7. A planet has a semi-major axis of 4 AU. Use Kepler's Third Law to calculate its orbital period in Earth years.
8. A planet takes 64 years to orbit the Sun. Calculate its semi-major axis in AU.
9. Mars has a semi-major axis of approximately 1.52 AU. Use Kepler's Third Law to estimate its orbital period.
10. Satellite A orbits a planet with semi-major axis a and period T. Satellite B orbits the same planet with semi-major axis 4a. Determine Satellite B's orbital period in terms of T.
11. Explain how Newton's law of universal gravitation provides a physical explanation for Kepler's Third Law.
12. Why does the mass of a small satellite not appear in the simplified equation T² = 4π²a³/(GM)?
13. Explain how astronomers can use the orbit of a planet to estimate the mass of its star.
14. Describe how each of Kepler's three laws is relevant to an interplanetary spacecraft travelling along an elliptical transfer orbit.
15. Astronomers discover three planets around the same star. Their measured values are:

Calculate or compare T² and a³ for each planet and evaluate whether the data are consistent with Kepler's Third Law.

Practical Applications of Orbital Mechanics

Orbital mechanics is the study of how objects move under the influence of gravity. It combines ideas from Newton's laws, gravitational fields, energy, momentum, circular motion, and Kepler's laws to predict and control the motion of satellites and spacecraft.

Orbital mechanics is not only important for space exploration. Modern society relies on satellites for:

• communication
• navigation
• weather forecasting
• Earth observation
• climate monitoring
• scientific research
• disaster response
• mapping
• transportation
• timing systems

For every satellite mission, engineers must decide where the satellite should orbit, how it will get there, and how its orbit will be maintained.

---

From Physics to Engineering

The fundamental equations of orbital mechanics allow engineers to predict satellite motion.

For a circular orbit:

v = √(GM/r)

Orbital period is:

T = 2π√(r³/GM)

Gravitational field strength is:

g = GM/r²

Circular-orbit total energy is:

E_total = −GMm/(2r)

These relationships allow engineers to calculate:

• orbital speed
• orbital period
• required altitude
• gravitational acceleration
• orbital energy
• transfer trajectories
• escape conditions

The equations become practical engineering tools when designing satellite missions.

---

Choosing the Correct Orbit

There is no single best orbit for every satellite.

Mission planners first ask:

What must the satellite accomplish?

They then select an orbit with appropriate:

• altitude
• inclination
• eccentricity
• orbital period
• orientation
• coverage pattern

A communication satellite may require a very different orbit from an imaging satellite.

---

Communication Satellites

Satellites allow signals to travel between distant locations on Earth.

A communication satellite may receive a signal from one ground station, process or amplify it, and transmit it toward another location.

Satellite communications are used for:

• television
• internet
• telephone services
• maritime communication
• aircraft communication
• emergency communication
• military communication
• remote communities

The choice of orbit strongly affects how the communication system operates.

---

Geostationary Communication Satellites

A geostationary satellite appears approximately stationary above one longitude on Earth's equator.

Its altitude is about:

35,786 km

and its orbital period matches Earth's rotation.

This provides a major advantage:

ground antennas can point continuously toward the same location in the sky.

The antenna does not need to track a rapidly moving satellite.

---

Advantages of GEO for Communication

Geostationary satellites provide:

• continuous coverage of the same large region
• very large coverage areas
• predictable satellite position
• simple fixed ground antennas
• continuous television and telecommunications services

A small number of GEO satellites can provide broad coverage across much of Earth.

---

Limitations of GEO

The large distance creates disadvantages.

Signals must travel tens of thousands of kilometres to the satellite and back.

Since electromagnetic waves travel at approximately:

c = 3.0 × 10⁸ m/s

there is a noticeable propagation delay.

GEO satellites also provide poor viewing geometry at very high latitudes.

---

LEO Communication Satellites

Communication systems can instead use satellites in Low Earth Orbit (LEO).

These satellites may orbit only hundreds or around a thousand kilometres above Earth.

Because they are closer:

• signal travel distance is shorter
• latency can be lower
• less transmitter power may be required

However, each satellite sees a smaller portion of Earth.

---

Why LEO Communication Requires Constellations

A LEO satellite moves quickly relative to Earth's surface.

It may only remain visible from a particular location for a short time.

Continuous service therefore requires multiple satellites arranged in a:

satellite constellation

As one satellite disappears below the horizon, communication can be transferred to another.

This is called a:

handover

Orbital mechanics is used to determine how many satellites are required and how they should be spaced.

---

Navigation Satellites

Satellite navigation systems allow receivers to determine their position on Earth.

Examples include global navigation satellite systems used by:

• smartphones
• aircraft
• ships
• cars
• surveying equipment
• emergency services
• scientific instruments

Navigation satellites broadcast extremely precise information about:

• satellite position
• transmission time

A receiver compares signals from several satellites to determine its location.

---

Why Several Satellites Are Needed

A receiver determines its position by measuring signal travel times from multiple satellites.

Because:

distance = speed × time

and radio signals travel at approximately the speed of light, precise timing provides distance information.

Signals from several satellites allow the receiver to determine:

• latitude
• longitude
• altitude
• receiver clock correction

Therefore, navigation systems require carefully designed constellations.

---

Medium Earth Orbit for Navigation

Many navigation satellites operate in Medium Earth Orbit (MEO).

MEO provides a useful balance between:

• large coverage area
• manageable signal strength
• long satellite visibility
• constellation size

A higher satellite sees more of Earth than a LEO satellite, so fewer satellites can provide broad coverage.

---

Timing Is as Important as Position

Navigation satellites carry extremely precise clocks.

Even a tiny timing error can produce a large position error.

For example, light travels approximately:

300,000 km/s

or:

300 m in one microsecond

Therefore, a timing error of only:

1 μs

corresponds to roughly:

300 m

of signal travel distance.

This shows why satellite navigation depends on extraordinarily accurate timing.

---

Earth Observation

Satellites provide a powerful way to observe Earth repeatedly from space.

Earth-observation satellites can monitor:

• forests
• crops
• cities
• oceans
• glaciers
• sea ice
• volcanoes
• wildfires
• floods
• drought
• air pollution
• land-use change

---

Why LEO Is Useful for Earth Observation

Many imaging satellites use LEO because the satellite is relatively close to Earth's surface.

This can provide high spatial resolution.

A lower orbit can allow instruments to detect smaller surface features than an otherwise similar instrument much farther away.

However, a single LEO satellite cannot continuously observe the same location.

---

Polar Orbits

Many Earth-observation satellites use polar or near-polar orbits.

As the satellite travels north and south, Earth rotates beneath it.

Each orbit therefore passes over a different strip of Earth's surface.

Over time, the satellite can observe nearly the entire planet.

---

Sun-Synchronous Orbits

Many Earth-imaging satellites use a special near-polar orbit called a:

Sun-synchronous orbit

The orbital plane slowly changes orientation so that the satellite crosses a particular region at approximately the same local solar time on repeated passes.

This provides similar lighting conditions.

That is extremely useful when scientists want to compare images taken on different days or years.

---

Example: Monitoring Forest Change

Suppose scientists want to measure deforestation.

If one image is taken early in the morning and another near sunset, different shadows could make comparison difficult.

A Sun-synchronous orbit can provide observations under more consistent lighting.

Changes in the images are then more likely to represent actual changes on the ground rather than changes in illumination.

---

Weather Satellites

Weather forecasting uses satellites in several different orbits.

Geostationary weather satellites provide continuous views of large regions.

Polar-orbiting weather satellites provide detailed observations across the entire planet.

These orbit types complement each other.

---

GEO Weather Monitoring

A geostationary satellite can repeatedly image the same region every few minutes.

This is particularly useful for monitoring:

• tropical cyclones
• thunderstorms
• cloud movement
• developing storms

Scientists can observe how weather systems change over time.

---

Polar Weather Monitoring

Polar satellites provide closer and often more detailed observations.

They can also observe high latitudes that GEO satellites cannot view well.

Therefore, modern weather forecasting benefits from using multiple orbit types rather than relying on one.

---

Disaster Response

Satellite observations are particularly valuable when disasters affect large or inaccessible areas.

Satellites can help monitor:

• hurricanes
• floods
• wildfires
• volcanic eruptions
• landslides
• earthquakes and their effects
• oil spills

Repeated satellite passes allow emergency agencies to compare conditions before and after an event.

---

Satellite Mapping

Satellites contribute to modern mapping by providing repeated observations of Earth's surface.

Orbital mechanics determines:

• when the satellite passes over an area
• the angle from which it observes
• how frequently it returns
• how much surface it can see

This leads to the important idea of revisit time.

---

Revisit Time

Revisit time is the time between observations of the same location.

A mission monitoring rapidly changing events may need a short revisit time.

For example:

wildfire monitoring → short revisit time desirable

while:

long-term geological mapping → longer revisit time may be acceptable

Satellite constellations can reduce revisit time because several satellites can observe the same region.

---

Coverage Versus Resolution

Orbit selection often involves a trade-off between:

coverage

and:

resolution

A high satellite can see a large area of Earth.

A low satellite can usually observe smaller details more easily.

Therefore:

higher altitude → generally greater coverage

lower altitude → potentially greater spatial resolution

Engineers must choose the balance appropriate to the mission.

---

Orbital Inclination

Another important orbital property is inclination.

Inclination describes the angle between the satellite's orbital plane and Earth's equatorial plane.

Inclination affects which parts of Earth the satellite can observe.

For example:

• low inclination → mainly low latitudes
• high inclination → higher latitudes accessible
• near 90° → near-polar coverage

Therefore, inclination is critical for Earth-observation mission design.

---

Launch Site and Inclination

Earth's rotation can help launch a satellite.

A launch toward the east can take advantage of Earth's rotational velocity.

This effectively provides some initial speed without requiring additional rocket propellant.

The benefit is greatest near the equator because Earth's surface moves fastest there.

However, the required final inclination may determine whether this advantage can be fully used.

---

Maintaining Satellite Orbits

A real satellite does not remain forever in a perfectly ideal Keplerian orbit.

Many small influences gradually alter its trajectory.

These are called:

orbital perturbations

Engineers must predict and sometimes correct them.

---

Atmospheric Drag

Satellites in LEO encounter traces of Earth's upper atmosphere.

Even at hundreds of kilometres altitude, a small number of atmospheric particles remain.

Collisions with these particles produce drag.

Drag removes orbital energy.

Therefore:

E_total becomes more negative

and the satellite's orbit gradually becomes lower.

This process is called:

orbital decay.

---

Increasing Drag as the Orbit Drops

As the satellite moves lower:

• atmospheric density increases
• drag increases
• energy loss increases
• orbital decay accelerates

Eventually, a satellite may re-enter Earth's atmosphere unless its orbit is raised.

---

Orbital Reboost

Some spacecraft use engines to periodically increase their orbital energy.

This is called an orbital reboost.

A forward engine burn can:

• increase velocity
• increase orbital energy
• raise part of the orbit

Further manoeuvres can restore the desired orbit.

---

Earth's Shape Affects Orbits

Earth is not a perfect sphere.

It bulges slightly around the equator.

Therefore, Earth's gravitational field is not perfectly symmetrical.

This causes satellite orbital planes and orbital orientations to gradually shift.

These effects are sometimes unwanted.

However, engineers can also use them deliberately.

---

Sun-Synchronous Orbits Use a Perturbation

Sun-synchronous satellites take advantage of the effect of Earth's equatorial bulge.

The orbit is chosen so that Earth's non-spherical gravitational field causes the orbital plane to slowly rotate at approximately the required rate.

This keeps the satellite's observation time relative to the Sun nearly constant.

This is an excellent example of engineers using an orbital perturbation rather than simply fighting it.

---

Gravitational Perturbations

Other celestial bodies also affect satellite orbits.

A satellite around Earth may experience small gravitational influences from:

• Moon
• Sun
• planets

For many low satellites these effects are relatively small, but they can become important for:

• high-altitude satellites
• long-duration missions
• extremely precise navigation

---

Solar Radiation Pressure

Light carries momentum.

When sunlight strikes a spacecraft, it exerts a tiny force called:

solar radiation pressure

For a massive satellite the effect may be small, but over long periods it can gradually alter an orbit.

The effect is particularly important for spacecraft with:

• large solar panels
• large reflective surfaces
• very low mass relative to area

---

Station-Keeping

Station-keeping refers to manoeuvres used to keep a satellite within its required orbital region.

Small engine burns can correct changes in:

• altitude
• eccentricity
• inclination
• orbital orientation
• longitude

For geostationary satellites, station-keeping is especially important because the satellite must remain near its assigned orbital position.

---

Propellant Is Limited

Traditional station-keeping uses spacecraft propulsion.

Every engine burn consumes propellant.

A satellite has only a limited supply.

Therefore:

orbital maintenance can limit mission lifetime.

Engineers carefully design manoeuvres to minimize unnecessary propellant use.

Electric propulsion can also be used on some satellites because it provides very efficient, although relatively low, thrust.

---

Collision Avoidance

Earth orbit contains many:

• active satellites
• inactive satellites
• rocket bodies
• fragments
• pieces of orbital debris

At orbital speeds, even a small object can carry considerable kinetic energy.

Therefore, satellite operators track objects and estimate collision probabilities.

---

Collision-Avoidance Manoeuvres

If another object is predicted to pass dangerously close, operators may perform a:

collision-avoidance manoeuvre

The satellite's velocity is changed slightly.

Even a small velocity change made sufficiently early can produce a large separation by the predicted encounter time.

This is another direct application of orbital mechanics.

---

Why Small Changes Matter

Suppose a satellite changes its velocity by only a small amount.

Immediately, the difference in position may be tiny.

But over hours or days, the satellite follows a slightly different orbit.

The separation between the predicted trajectories can become many kilometres.

Therefore:

small Δv applied early can create large future position changes.

---

Orbital Debris

A collision between satellites can produce thousands of fragments.

Each fragment can become another collision hazard.

This creates concern about cascading collisions in heavily populated orbital regions.

Reducing orbital debris is therefore an important engineering and environmental challenge.

---

End-of-Life Planning

Satellite missions should include plans for what happens when the satellite stops operating.

Depending on the orbit, engineers may:

• deliberately deorbit the satellite
• move it to a lower orbit that will decay
• move it to a disposal orbit
• remove stored energy from batteries and pressure systems

Orbital mechanics is therefore important even after the useful mission has ended.

---

Disposal Orbits

High-altitude satellites cannot always be practically deorbited.

For some geostationary satellites, operators can raise them into a disposal orbit above the heavily used GEO region.

This reduces interference with operational spacecraft.

Such an orbit is sometimes called a graveyard orbit.

---

Orbital Transfers

Satellites frequently need to move from one orbit to another.

This requires changing:

• velocity
• orbital energy
• sometimes inclination

One efficient transfer between two approximately circular coplanar orbits is the:

Hohmann transfer.

---

Raising a Satellite's Orbit

To move from a lower circular orbit to a higher circular orbit:

1. the satellite performs a forward burn
2. kinetic and total orbital energy increase
3. the satellite enters an elliptical transfer orbit
4. it coasts toward the higher altitude
5. another burn establishes the new circular orbit

The process uses orbital mechanics to minimize required propulsion.

---

Plane Changes

Sometimes a satellite must change the orientation of its orbital plane.

This is called a:

plane change

Plane changes require changing the direction of the spacecraft's velocity.

They can require substantial Δv, especially when the spacecraft is moving quickly.

Therefore, mission designers usually try to launch satellites close to their required orbital inclination rather than making large plane changes later.

---

Why Plane Changes Can Be Expensive

Velocity is a vector.

Changing its direction requires a change in velocity even if speed remains approximately constant.

If the original and final velocity vectors differ greatly, the required:

Δv

can be large.

Large Δv generally means more propellant.

---

Rendezvous and Docking

Two spacecraft cannot rendezvous simply by pointing one directly at the other and accelerating.

Both spacecraft are moving in orbit.

To meet safely, their:

• orbital planes
• altitudes
• orbital periods
• positions
• velocities

must be carefully coordinated.

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Catching Another Spacecraft

A counterintuitive feature of orbital mechanics is that a spacecraft may need to enter a lower orbit to catch up with another spacecraft.

Lower circular orbits have:

• greater orbital speed
• shorter orbital periods

Therefore, a spacecraft in a lower orbit moves around Earth more quickly and can change its relative position.

This technique is called orbital phasing.

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Space Stations

Crewed space stations depend heavily on orbital mechanics.

Engineers must manage:

• launch trajectories
• rendezvous
• docking
• atmospheric drag
• reboosts
• collision avoidance
• visiting spacecraft
• eventual deorbiting

The station and approaching spacecraft may both be travelling at roughly several kilometres per second.

Successful docking requires their relative velocity to become extremely small.

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Interplanetary Travel

Orbital mechanics also governs spacecraft travelling between planets.

A spacecraft leaving Earth does not simply fly through stationary space toward a stationary destination.

Both Earth and the destination planet are moving around the Sun.

Mission planners must predict where the destination planet will be when the spacecraft arrives.

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Launch Windows

A launch window is a period when planetary positions allow a suitable trajectory.

For an interplanetary mission:

• Earth must be correctly positioned
• destination planet must be correctly positioned
• spacecraft travel time must match the changing geometry

Launching at the wrong time could require much more energy or make the desired trajectory impossible.

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Transfer Orbits Between Planets

Interplanetary spacecraft commonly travel along elliptical heliocentric trajectories.

For example, a simplified Earth-to-Mars transfer uses part of an ellipse whose:

• perihelion is near Earth's orbit
• aphelion is near Mars's orbit

The spacecraft follows this trajectory because of the Sun's gravity.

Rocket engines establish the required initial trajectory, but much of the journey is an unpowered gravitational coast.

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Gravity Assists

Orbital mechanics allows spacecraft to use planets to alter their trajectories.

During a gravity assist, a spacecraft passes close to a moving planet.

The planet's gravity bends the spacecraft's velocity vector.

Relative to the Sun, this can change the spacecraft's:

• speed
• direction
• orbital energy
• angular momentum

Gravity assists have made many ambitious planetary missions possible while reducing propulsion requirements.

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Planetary Orbit Insertion

Reaching another planet is only part of the problem.

If a spacecraft approaches with positive orbital energy relative to the planet, it will simply fly past.

To enter orbit, energy must be removed.

Possible methods include:

• rocket engine burn
• atmospheric braking
• combinations of manoeuvres

Once:

E_total < 0

relative to the planet, the spacecraft becomes gravitationally bound.

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Aerobraking

Some spacecraft can use a planet's atmosphere to gradually reduce orbital energy.

During aerobraking:

• atmospheric drag acts on the spacecraft
• mechanical energy decreases
• some energy becomes thermal energy
• the orbit gradually changes

This can reduce the amount of propellant required.

However, engineers must carefully control heating and atmospheric forces.

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Landing on Other Worlds

Orbital mechanics also affects landing.

Before landing, a spacecraft may need to remove enormous amounts of:

• orbital kinetic energy
• gravitational potential energy

Different worlds require different strategies.

A body with an atmosphere may allow:

• heat shields
• parachutes
• aerodynamic braking

A body without a useful atmosphere may require:

• rocket propulsion
• carefully timed descent burns

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Returning to Earth

Returning spacecraft must also change their orbit.

A spacecraft in Earth orbit can perform a deorbit burn.

This lowers the orbit so that the trajectory intersects the atmosphere.

Atmospheric drag then removes much of the spacecraft's remaining mechanical energy.

The challenge is controlling:

• re-entry angle
• heating
• deceleration
• landing location

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Modern Space Technology

Orbital mechanics supports technologies used every day.

Examples include:

Navigation

Satellite positioning guides vehicles, ships, aircraft and smartphones.

Communication

Satellites connect distant regions and support broadcasting and internet services.

Weather

Satellites track clouds, storms and atmospheric conditions.

Agriculture

Remote sensing helps monitor crops, soil moisture and vegetation.

Climate science

Satellites measure ice, oceans, atmospheric gases and temperature changes.

Disaster management

Satellites monitor floods, fires, storms and other hazards.

Scientific research

Space telescopes and scientific satellites study Earth and the universe.

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Mission Requirements and Orbit Choice

Consider several example missions.

Mission A: Continuous Regional Weather Monitoring

Requirements:

• constant view of one region
• broad coverage
• repeated imaging

Suitable orbit:

GEO

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Mission B: Detailed Global Mapping

Requirements:

• high-resolution images
• global coverage
• consistent lighting

Suitable orbit:

Sun-synchronous LEO

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Mission C: Global Navigation

Requirements:

• several satellites visible simultaneously
• broad coverage
• predictable geometry

Suitable orbit:

MEO constellation

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Mission D: Low-Latency Global Communication

Requirements:

• relatively short signal distance
• worldwide coverage

Possible solution:

large LEO constellation

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Mission E: Continuous High-Latitude Communication

Requirement:

• long visibility over high northern or southern regions

Possible solution:

highly elliptical orbit or suitable constellation

The orbit is selected according to the mission—not the other way around.

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Engineering Trade-Offs

Satellite design often requires balancing competing goals.

Improving one feature may make another worse.

This is why orbital design is an optimization problem.

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Maintaining an Accurate Orbit

Some missions require extremely accurate knowledge of satellite position.

Navigation satellites are an obvious example.

Engineers monitor satellite trajectories using:

• ground tracking
• radio signals
• ranging
• onboard sensors

Computer models then predict future positions.

Small corrections can be performed when necessary.

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Challenges in Satellite Operations

Maintaining a useful satellite orbit requires engineers to manage many problems:

• atmospheric drag
• gravitational perturbations
• solar radiation pressure
• orbital debris
• limited propellant
• equipment failures
• radiation
• thermal cycling
• navigation errors
• communication delays
• space weather

A successful mission requires far more than simply reaching orbit.

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Space Weather

Activity from the Sun can influence satellite operations.

Solar activity can:

• disturb Earth's magnetic environment
• affect electronics
• disrupt communication
• increase radiation exposure

It can also heat and expand Earth's upper atmosphere.

For low-orbit satellites, this can increase atmospheric drag and accelerate orbital decay.

Thus even solar activity can indirectly affect orbital mechanics.

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Increasing Numbers of Satellites

The number of satellites in Earth orbit has increased substantially.

This creates benefits such as:

• improved communications
• more frequent Earth imaging
• better global services

but also challenges:

• collision risk
• radio interference
• orbital congestion
• debris
• end-of-life disposal

Orbital mechanics is increasingly important for managing Earth orbit as a shared environment.

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Autonomous Satellite Operations

Modern spacecraft can perform some orbital tasks automatically.

Onboard computers may help:

• determine position
• control orientation
• execute engine burns
• maintain formation
• respond to faults

Future spacecraft may rely increasingly on autonomous navigation for missions far from Earth, where communication delays make immediate ground control impossible.

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Formation Flying

Some missions use several spacecraft flying in carefully controlled relative positions.

This is called formation flying.

The satellites may work together as if they were parts of one much larger instrument.

Maintaining the formation requires precise understanding of:

• relative orbital motion
• gravitational differences
• perturbations
• small velocity corrections

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Orbital Mechanics and Space Telescopes

Not every spacecraft operates in a simple Earth orbit.

Some observatories are placed in special locations or trajectories where gravitational effects from multiple bodies are important.

Mission planners may choose these locations because they provide:

• stable observing conditions
• useful thermal environments
• favourable communication geometry
• reduced obstruction from Earth

This shows that orbital mechanics extends beyond simple circular satellite motion.

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Evaluating a Satellite Mission

When evaluating an orbital design, engineers might ask:

• Does the orbit provide the required coverage?
• Is the revisit time acceptable?
• Is communication delay acceptable?
• Can the launch vehicle reach the orbit?
• How much propellant is required?
• How much station-keeping will be needed?
• What perturbations will affect the orbit?
• Is collision risk manageable?
• How long can the satellite operate?
• What is the end-of-life plan?

A technically possible orbit is not necessarily the best operational orbit.

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A Complete Mission Example

Imagine engineers are designing a satellite to monitor global crop health.

The satellite must:

• observe agricultural regions worldwide
• detect relatively small surface features
• compare changes over months and years
• maintain similar lighting between observations

A suitable choice would be:

near-polar Sun-synchronous LEO

Why?

LEO

provides relatively high-resolution observations.

Near-polar inclination

provides global coverage as Earth rotates beneath the orbit.

Sun-synchronous geometry

provides approximately consistent local observation times.

However, the satellite must also deal with:

• atmospheric drag
• limited visibility from individual ground stations
• orbital debris
• possible orbit corrections

The mission design therefore combines scientific goals with orbital engineering.

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Common Mistakes

Thinking satellites remain in orbit without gravity

Gravity provides the inward acceleration that produces orbital motion.

Thinking one orbit is best for every satellite

Different mission objectives require different orbits.

Thinking GEO is ideal for all communication

GEO provides excellent continuous regional coverage but has greater latency and poor polar geometry.

Thinking polar and LEO mean the same thing

LEO describes altitude; polar describes orbital inclination.

Thinking satellite orbits never change

Real orbits are affected by drag and other perturbations.

Thinking station-keeping is unnecessary

Many satellites require periodic corrections.

Thinking a spacecraft can simply point toward another spacecraft to rendezvous

Orbital period, velocity and relative motion must be considered.

Thinking a spacecraft travels directly toward where a planet is now

Interplanetary trajectories must target where the planet will be at arrival.

Thinking small engine burns have little effect

A small Δv can create a large positional difference over time.

Ignoring satellite disposal

End-of-life planning is an important part of responsible satellite operations.

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Key Terms

Orbital mechanics — study of the motion of objects under gravitational forces.

Satellite constellation — coordinated group of satellites operating together.

Low Earth Orbit (LEO) — relatively low-altitude orbit used for many imaging, scientific and communication missions.

Medium Earth Orbit (MEO) — region between LEO and GEO, widely used for navigation systems.

Geostationary Earth Orbit (GEO) — equatorial orbit in which a satellite appears approximately stationary above one longitude.

Polar orbit — highly inclined orbit passing near Earth's poles.

Sun-synchronous orbit — near-polar orbit designed to cross locations at approximately the same local solar time.

Revisit time — time between repeated observations of the same location.

Orbital inclination — angle between an orbital plane and a reference plane such as Earth's equator.

Orbital perturbation — deviation from an ideal orbit caused by additional forces or gravitational influences.

Orbital decay — gradual reduction of orbital energy and altitude.

Station-keeping — manoeuvres used to maintain a satellite's desired orbit.

Reboost — manoeuvre that increases orbital energy and raises an orbit.

Collision-avoidance manoeuvre — controlled change in orbit used to reduce collision risk.

Orbital debris — non-functioning human-made objects or fragments remaining in orbit.

Delta-v — change in velocity required for a spacecraft manoeuvre.

Hohmann transfer — efficient elliptical transfer between two approximately circular coplanar orbits.

Plane change — manoeuvre changing the orientation of an orbital plane.

Orbital phasing — adjustment of orbital period to change relative position between spacecraft.

Rendezvous — controlled meeting of spacecraft in orbit.

Launch window — period when orbital geometry allows a desired mission trajectory.

Orbit insertion — manoeuvre used to place a spacecraft into a bound orbit.

Aerobraking — use of atmospheric drag to reduce orbital energy.

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Key Takeaways

• Orbital mechanics turns gravitational physics into practical spacecraft engineering.
• Satellite orbit selection depends on mission objectives.
• Communication, navigation, weather, Earth observation and scientific missions require different orbital characteristics.
• GEO provides continuous coverage of approximately the same region.
• LEO offers shorter communication distances and can provide high-resolution Earth observations.
• MEO is widely used for navigation satellite constellations.
• Polar orbits provide global coverage over time.
• Sun-synchronous orbits provide consistent lighting for repeated Earth observations.
• Satellite constellations can provide continuous or frequent global coverage.
• Orbital mechanics determines coverage, revisit time, speed, period and communication geometry.
• Real satellite orbits are affected by perturbations.
• Atmospheric drag removes orbital energy from low satellites.
• Station-keeping and reboost manoeuvres can maintain desired orbits.
• Small velocity changes can create large trajectory changes over time.
• Collision avoidance is increasingly important as Earth orbit becomes more crowded.
• Orbital transfers allow satellites to move efficiently between different orbits.
• Rendezvous requires matching orbital position and velocity rather than simply flying directly toward another spacecraft.
• Interplanetary missions depend on launch windows and transfer trajectories.
• Gravity assists can change spacecraft velocity and orbital energy while reducing propulsion requirements.
• Orbit insertion, aerobraking and landing all require careful management of orbital energy.
• Orbital mechanics supports technologies used daily for communication, navigation, weather forecasting, mapping and environmental monitoring.
• Modern space exploration is fundamentally an application of gravitational fields, energy, Newton's laws and Kepler's laws.

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Check Your Understanding

1. Define orbital mechanics and explain why it is important for satellite operations.
2. Explain why a communication satellite might be placed in GEO rather than LEO.
3. Give two advantages and two disadvantages of using LEO for satellite communication.
4. Explain why navigation systems require constellations of satellites rather than a single satellite.
5. A timing error of 2 μs occurs in a navigation signal. Using c = 3.0 × 10⁸ m/s, calculate the corresponding error in signal travel distance.
6. Explain why a near-polar Sun-synchronous orbit is useful for monitoring changes in forests, crops or glaciers.
7. Compare GEO and polar orbits for weather observation. Explain one advantage of each.
8. Explain how atmospheric drag changes the orbital energy and altitude of a LEO satellite.
9. Describe what is meant by station-keeping and explain why a satellite may require it.
10. Explain why a small collision-avoidance Δv applied several hours before an encounter can significantly change the satellite's later position.
11. Why are large orbital plane changes usually avoided when possible?
12. Explain why a spacecraft trying to catch another spacecraft may first enter a lower orbit rather than accelerating directly toward it.
13. Describe how orbital mechanics determines the launch window for an Earth-to-Mars mission.
14. Design an appropriate orbit for a satellite whose mission is to produce repeated high-resolution images of the entire Earth under similar lighting conditions. Justify your choices of altitude range, inclination and orbit type.
15. A company wants one satellite design to provide low-latency internet, continuous hurricane monitoring, high-resolution global mapping, and precise global navigation. Evaluate why a single orbit is unlikely to perform all four jobs optimally, and recommend suitable orbital approaches for each application.