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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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Perihelion Versus Aphelion

So:

closer to Sun → faster

farther from Sun → slower

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

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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

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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.

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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.

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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.

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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.

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Planetary Data

Approximate Solar System values illustrate Kepler's Third Law:

As orbital size increases, orbital period increases strongly.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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Useful Scaling Patterns

Since:

T ∝ a^(3/2)

we obtain:

This relationship is much stronger than a simple direct proportion.

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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.

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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

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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)

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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.

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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.

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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.

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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.