# Energy Changes Within and Between Orbits

This topic is part of the HSC Physics syllabus under the section Motion in Gravitational Fields.

### HSC Physics Syllabus

• Derive quantitatively and apply the concepts of gravitational force and gravitational potential energy in radial gravitational fields to a variety of situations, including but not limited to:

– total potential energy of a planet or satellite in its orbit U=-(GMm)/r
– total energy of a planet or satellite in its orbit U+K=-(GMm)/(2r)
– energy changes that occur when satellites move between orbits
– Kepler’s Laws of Planetary Motion

### Gravitational Potential Energy in Orbit

Gravitational potential energy of an object in orbit is determined by its mass and orbital radius (distance from a large mass).

Gravitational potential energy is given by:

$$U = -\frac{GMm}{r}$$

Gravitational potential energy changes with orbital radius in an elliptical orbit. It increases (becomes less negative) as orbital radius increases; decreases (becomes more negative) as orbital radius decreases.

In contrast, gravitational potential energy of an object remains constant in a circular orbit due to a fixed orbital radius.

### Kinetic Energy in Orbit

Besides gravitational potential energy, all objects in orbital motion possess kinetic energy given by:

K=1/2mv^2

#### Circular Orbits

In a stable circular orbit, the orbital velocity of an object remains constant. Orbital velocity in a circular orbit is given by:

$$v_{\text{orbital}} = \sqrt{\frac{GM}{r}}$$

By substituting orbital velocity into kinetic energy equation:

K=1/2m(sqrt((GM)/r))^2

K=(GMm)/(2r)

#### Elliptical Orbits

For elliptical orbits such as those of planets around stars, the orbital velocity of an object changes with orbital radius. It decreases as the orbital radius increases, and increases as orbital radius decreases. Thus, the kinetic energy of these objects also decreases with increasing orbital radius.

### Total Mechanical Energy in Orbit

Total mechanical energy (TE) of an object in orbit equals to sum of its kinetic energy (K) and gravitational potential energy (U):

TE = K + U
For circular orbits:

TE = (GMm)/(2r) – (GMm)/r

TE = - (GMm)/(2r)

TE = 1/2U
Therefore, total energy of an object in a circular orbit is exactly half of its gravitational potential energy (see graph).

The total energy of an object in an elliptical orbit is given by:

$$TE = -\frac{GMm}{2a}$$

where a is the semi-major axis of the elliptical orbit.

### Change in Energy in Elliptical Orbit

• Mechanical Energy: Total energy remains constant and the law of conservation of mechanical energy applies due to the absence of any non-conservative forces. If the propeller or thrust of a spacecraft or satellite is turned off, the mechanical energy remains constant throughout its orbit. This also applies to orbital motion of planets.

• Gravitational Potential Energy: As orbital radius r decreases, an orbiting object's gravitational potential energy decreases (becomes more negative) and is transformed into its kinetic energy. Vice versa, As r increases, gravitational potential energy increases (becomes less negative).
• Kinetic Energy: Changes in kinetic energy can be accounted by changes in an object's gravitational potential energy. When orbital radius increases, kinetic energy is transformed into gravitational potential energy. Conversely, when orbital radius decreases, gravitational potential energy is transformed into kinetic energy.

These changes in energy are consistent with Kepler's second of planetary motion which states that an imaginary line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This law implies that planets move faster when they are closer to the Sun (greater kinetic energy) and slower when they are farther away (smaller kinetic energy).

### Changes in Energy Between Orbits

The total mechanical energy of an object changes depending on the radius of orbit (shown in graph above).

• In orbits with greater radius or semi-major axis, the total energy is greater (smaller kinetic energy and greater gravitational potential energy).
• In orbits with smaller radius or semi-major axis, the total energy is smaller (greater kinetic energy and smaller gravitational potential energy).

Previous section: Escape Velocity

Next section: Low Earth and Geostationary Orbit Satellites