1. Introduction

Planets remain in orbit around the Sun because gravity continuously pulls them toward the Sun while their forward motion carries them around it.

An orbit is a continuous state of free fall in which a planet is constantly falling toward the Sun but never collides with it.

The study of planetary orbits combines:

• Newton’s Law of Gravitation
• Circular Motion
• Energy Conservation
• Angular Momentum Conservation
• Kepler’s Laws

2. Why Planets Orbit the Sun

The Sun exerts a gravitational force on every planet.

According to Newton’s Law of Gravitation:

FG=GMSmr2F_G = \frac{GM_Sm}{r^2}FG​=r2GMS​m​

where:

• $G$ = universal gravitational constant
• MSM_SMS​ = mass of the Sun
• $m$ = mass of the planet
• $r$ = distance between centers

This gravitational force acts as the centripetal force required for orbital motion.

3. Gravity as a Centripetal Force

For circular motion:

FC=mv2rF_C = \frac{mv^2}{r}FC​=rmv2​

For a planet in orbit:

FG=FCF_G = F_CFG​=FC​

Therefore:

GMSmr2=mv2r\frac{GM_Sm}{r^2} = \frac{mv^2}{r}r2GMS​m​=rmv2​

The planet’s mass cancels.

4. Orbital Speed

Solving for orbital speed:

v=GMSrv = \sqrt{\frac{GM_S}{r}}v=rGMS​​​

Important Observations

The orbital speed depends on:

• the mass of the Sun
• the orbital radius

The orbital speed does not depend on the planet’s mass.

5. Relationship Between Radius and Speed

Because:

v=GMSrv = \sqrt{\frac{GM_S}{r}}v=rGMS​​​

a larger orbital radius produces a smaller orbital speed.

Consequence

Inner planets move faster.

Outer planets move slower.

For example:

• Mercury travels much faster than Neptune.
• Venus travels faster than Saturn.

6. Orbital Period

The orbital period is the time required to complete one orbit.

For circular motion:

v=2πrTv = \frac{2\pi r}{T}v=T2πr​

where:

• $T$ = orbital period

Substituting the orbital-speed equation gives:

T=2πr3GMST = 2\pi \sqrt{\frac{r^3}{GM_S}}T=2πGMS​r3​​

7. Kepler’s Third Law

Squaring the orbital-period equation:

T2=4π2GMSr3T^2 = \frac{4\pi^2}{GM_S} r^3T2=GMS​4π2​r3

This leads directly to Kepler’s Third Law:

T2∝r3T^2 \propto r^3T2∝r3

Planets farther from the Sun require longer periods to complete an orbit.

8. Elliptical Orbits

Real planetary orbits are not perfectly circular.

According to Kepler’s First Law:

Every planet moves in an ellipse with the Sun at one focus.

Characteristics of an Ellipse

An ellipse contains:

• a major axis
• a minor axis
• two foci

The Sun occupies one focus.

9. Perihelion and Aphelion

Perihelion

The point where a planet is closest to the Sun.

rmin⁡r_{\min}rmin​

At perihelion:

• gravitational force is greatest
• orbital speed is greatest

Aphelion

The point where a planet is farthest from the Sun.

rmax⁡r_{\max}rmax​

At aphelion:

• gravitational force is smallest
• orbital speed is smallest

10. Angular Momentum Conservation

The Sun’s gravitational force always points toward the Sun.

Therefore:

$$\tau =0$$

Since:

∑τ=dLdt\sum\tau = \frac{dL}{dt}∑τ=dtdL​

we obtain:

$$L=\text{constant}$$

Angular momentum remains conserved throughout the orbit.

11. Kepler’s Second Law

Because angular momentum is conserved:

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

This means:

• planets move faster near perihelion
• planets move slower near aphelion

12. Total Mechanical Energy of an Orbit

The total energy of a planet in orbit is:

$$E=K+U$$

where:

K=12mv2K=\frac12 mv^2K=21​mv2

and

U=−GMSmrU = -\frac{GM_Sm}{r}U=−rGMS​m​

13. Energy of a Circular Orbit

For a circular orbit:

K=GMSm2rK = \frac{GM_Sm}{2r}K=2rGMS​m​

Potential energy:

U=−GMSmrU = -\frac{GM_Sm}{r}U=−rGMS​m​

Therefore:

E=−GMSm2rE = -\frac{GM_Sm}{2r}E=−2rGMS​m​

Important Result

A bound orbit always has:

$$E<0$$

Negative total energy means the planet remains gravitationally bound to the Sun.

14. Escape from Orbit

To escape the Sun’s gravitational attraction completely:

$$E\ge 0$$

The planet or spacecraft must gain enough kinetic energy so that the total mechanical energy becomes nonnegative.

15. Example Problem

A planet orbits the Sun at:

$$r=4R$$

where $R$ is Earth’s orbital radius.

Compare its orbital period to Earth’s.

Solution

Using Kepler’s Third Law:

Tp2(4R)3=TE2R3\frac{T_p^2}{(4R)^3} = \frac{T_E^2}{R^3}(4R)3Tp2​​=R3TE2​​ Tp2=64TE2T_p^2 = 64T_E^2Tp2​=64TE2​ Tp=8TET_p = 8T_ETp​=8TE​

The planet requires eight times longer to complete one orbit.

16. Comparing Inner and Outer Planets

Inner Planets

Examples:

• Mercury
• Venus
• Earth
• Mars

Characteristics:

• smaller orbital radius
• shorter orbital period
• greater orbital speed

Outer Planets

Examples:

• Jupiter
• Saturn
• Uranus
• Neptune

Characteristics:

• larger orbital radius
• longer orbital period
• lower orbital speed

17. Common AP Physics C Mistakes

Mistake 1

Assuming planets move with constant speed.

Only circular orbits have constant speed.

Elliptical orbits do not.

Mistake 2

Using the planet’s mass in the orbital-speed equation.

v=GMSrv = \sqrt{\frac{GM_S}{r}}v=rGMS​​​

The planet’s mass cancels.

Mistake 3

Confusing orbital speed and orbital period.

A larger orbit means:

• lower speed
• longer period

18. Connections to Other Topics

Newton’s Law of Gravitation

Provides the force responsible for planetary motion.

Circular Motion

Gravity acts as the centripetal force.

Conservation of Energy

Explains orbital binding and escape.

Angular Momentum

Explains varying speeds in elliptical orbits.

Kepler’s Laws

Describe the observable behavior of planets.

Summary

Gravitational force:

FG=GMSmr2F_G = \frac{GM_Sm}{r^2}FG​=r2GMS​m​

Orbital speed:

v=GMSrv = \sqrt{\frac{GM_S}{r}}v=rGMS​​​

Orbital period:

T=2πr3GMST = 2\pi \sqrt{\frac{r^3}{GM_S}}T=2πGMS​r3​​

Total energy of a circular orbit:

E=−GMSm2rE = -\frac{GM_Sm}{2r}E=−2rGMS​m​

Kepler’s Third Law:

T2∝r3T^2 \propto r^3T2∝r3

Key ideas:

• planets remain in orbit because gravity provides the centripetal force
• inner planets move faster and have shorter periods
• outer planets move slower and have longer periods
• angular momentum and energy are conserved throughout orbital motion
• planetary orbits are elliptical, with the Sun located at one focus

The study of planetary orbits unifies gravitation, circular motion, energy, angular momentum, and Kepler’s Laws, making it one of the most important applications of AP Physics C Mechanics.