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Orbital Motion Relative to Other Object
by Ron Kurtus (revised 14 May 2011)
The orbital motion of two objects in space is often seen by an external observer as rotating about the center of mass (CM) or barycenter between them, as if the CM was a fixed axis-point.
However, to an observer on one of the objects, that object is fixed, while the other object appears to be orbiting the "fixed" object. A good example is how the Moon appears to orbit the Earth. An advantage of considering the orbit of one object around the other is that it simplifies the two orbital equations into one. This viewpoint is more convenient for calculating orbits and even escape velocity.
You can find the relative velocity of the one object as seen from the other by adding the two tangential velocity equations for circular orbits around the CM together and then rearranging the terms. A further simplification is seen when one object has a much greater mass than the other.
The equation for orbital velocity of one object with respect to the other is independent of which object is in orbit.
Questions you may have include:
- What are the tangential velocities of orbiting objects?
- What is the sum of their tangential velocities?
- What is an example of relative motion?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Tangential velocities of orbiting objects
The method to find the tangential velocity of one object with respect to the other is to start with the tangential velocities of the two objects with respect to the center of mass (CM) between them.
Velocities relative to CM
The tangential velocities of two objects in circular orbits around the center of mass (CM) between them and relative to that CM are in opposite directions, according to the relationship:
mvTm = −MvTM
The requirement for circular orbits around the CM is that the tangential velocities with are:
vTm = √[GM2/R(M + m)] km/s
vTM = √[Gm2/R(M + m)] km/s
where
- vTm is the tangential velocity of mass m in km/s
- vTM is the tangential velocity of mass M in km/s
- G is the Universal Gravitational Constant = 6.674*10−20 km3/kg-s2
- m is the mass of the smaller object in kg
- M is the mass of the larger object in kg
- R is the separation in kilometers (km) between the objects, as measured from their centers of mass
(See Derivation of Circular Orbits Around Center of Mass for more information.)
Velocity with respect to other object
If you shift the zero-point from the CM to the center of mass M, rM, the tangential velocity of mass m is the difference of the tangential velocities with respect to the CM.
vT = vTm − (−vTM)
Since the tangential velocities are in opposite directions, the magnitude of vT is simply the sum of the two speeds and its direction is the same as vTm. In other words, you can add the velocity equations.
The same logic holds if you consider the velocity of M with respect to m. In that case, vT = vTM − (−vTm). In other words, it does not matter which object is considered fixed.
Finding velocity relative to other object
You can find the tangential velocity of one object with respect to the other by adding their tangential velocities with respect to the CM:
vT = √[GM2/R(M + m)] + √[Gm2/R(M + m)]
Some algebraic manipulation is necessary:
vT = M√(G)/√[R(M + m)] + m√(G)/√[R(M + m)]
Combine both fractions over same denominator:
vT = [M√(G) + m√(G)]/√[R(M + m)]
vT = [(M + m)√(G)]/√[R(M + m)]
Note that (M + m) = √(M + m)2:
vT = √[G(M + m)2/R(M + m)]
Thus the tangential velocity for a circular orbit, as seen from the other object is:
vT= √[G(M + m)/R] km/s
In the case that M >> m (M is much greater than m), the equation reduces to:
vT= √(GM/R) km/s
Example of relative motion
An example of this relative motion is how the Moon appears to orbit the Earth. However, from the viewpoint of the Moon, the Earth appears to orbit the Moon at the same velocity.
Moon orbits the Earth
Considering the Earth has mass M and the Moon has mass m, you can see how the Moon appears to orbit the Earth in a counterclockwise direction:
Moon appears to orbit Earth
The CM between the objects is within the Earth's surface. That CM follows the Moon's orbit around the Earth.
Earth orbits the Moon
When astronauts were on the Moon, they saw the Earth orbiting the Moon.
Earth appears to orbit the Moon
Summary
The orbital motion of two objects in space is often seen by an external observer as rotating about the center of mass between them. However, to an observer on one of the objects, that object is fixed and the other object appears to be orbiting the "fixed" object.
You can find the relative velocity of the one object as seen from the other by adding the two tangential velocities for circular orbits around the CM together. The relative velocity is then the sum of the tangential velocities:
vT= √[G(M + m)/R] km/s
When M >> m, the equation becomes:
vT= √(GM/R) km/s
Shoot for the Moon
Resources and references
Websites
Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)
Center of Mass - Wikipedia
Books
Top-rated books on Simple Gravity Science
Top-rated books on Advanced Gravity Physics
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