Orbits of Planets

and Satellites

Orbits

An orbit is a repeating path that one object in space takes around another one:

An object in an orbit is called a satellite: it can be natural (Earth and moon) or man-made (machine that is launched into space).

Kepler’s Law

Kepler’s laws describe the motion of planets:

Although we usually apply the laws to planets orbiting the Sun, they hold equally well for satellites orbiting Earth or any other massive central body.

• 1^st Law of Orbit: All planets move in elliptical orbits, with the Sun at one focus.

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• 2^nd Law of Areas: The radial vector from the sun to a given planet sweeps out equal areas in equal time intervals.

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• 3^rd Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Kepler’s Law

1^st Law of Orbit: All planets move in elliptical orbits, with the Sun at one focus.

The eccentricity of Earth’s orbit is only 0.0167.

The Sun is at one of the two focal points.

Kepler’s Law

2^nd Law of Areas: The radial vector from the sun to a given planet sweeps out equal areas in equal time intervals.

Kepler’s Law

3^rd Law of Areas: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

If the orbit is circular:

Orbit speed:

Kepler’s Law

3^rd Law of Areas: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

If the orbit is circular:

and

Ex: Given the elliptical planetary orbit shown above, identify the interval during which the planet travels with the highest speed?

(A) Interval P1 to P2

(B) Interval P3 to P4

(C) They are the same.

The planet travels a greater distance from P1 to P2 in the same time interval.

Ex: A satellite orbits a planet in an elliptical path as shown. Specific positions of the satellite are noted on the diagram as A, B, C, and D.

Rank from highest to lowest the following characteristics of the satellite at each position:

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I) Speed

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II) Gravitational Potential Energy

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III) Total Mechanical Energy

A

B

C

D

Same (energy conservation)

Ex: A student is asked to determine the mass of Jupiter. Knowing which of the

following about Jupiter and one of its moons will allow the determination to be made?

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I. The time it takes for Jupiter to orbit the Sun

II. The time it takes for the moon to orbit Jupiter

III. The average distance between the moon and Jupiter

(A) I only

(B) II only

(C) III only

(D) I and II

(E) II and III

Kepler’s third law

Escape Speed

It’s the speed required for an object to completely escape the influence of gravity.

• If you fire a projectile upward, usually it will slow, stop momentarily, and return to Earth.

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• To get it far away from Earth, we want to bring its gravitational potential energy to zero.

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• But to find the minimum launch speed, we want the object’s final speed to be zero by the time it gets to this distant location (it has no leftover kinetic energy).

Escape Speed

It’s the speed required for an object to completely escape the influence of gravity.

By Conservation of Energy:

The gravitational pull by Jupiter provides the centripetal force on its moon:

Conservation of Angular Momentum

Consider a satellite that rotates around a planet in a elliptical orbit:

Angular momentum of a system is conserved if there is no net external torque acting on the system.

• At point A, the satellite is farther from the center of rotation than at point B.

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• Conservation of angular momentum tells us that the angular speed at point A must be less than at point B.

Conservation of Angular Momentum

Angular momentum of a system is conserved if there is no net external torque acting on the system.

Consider a satellite that rotates around a planet in a elliptical orbit:

a) What is the total energy of the asteroid?

Sun

Orbit of asteroid

Perihelion

aphelion

The total energy of the asteroid is equal to:

semimajor axis

Sun

Orbit of asteroid

Perihelion

aphelion

Applying Conservation of Angular Momentum:

At aphelion:

At perihelion:

c) What is the time necessary for the asteroid to make a complete orbit around the Sun?

Sun

Orbit of asteroid

Perihelion

aphelion