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Explanation of Effect of Velocity on Orbital Motion by Ron Kurtus - Succeed in Understanding Physics. Key words: tangential velocity, radial, center of mass, CM, elliptical, circular, parabolic, hyperbolic, physical science, School for Champions. Copyright © Restrictions
Effect of Velocity on Orbital Motion
by Ron Kurtus (revised 19 May 2011)
Although orbital motion of two objects in space can be with respect to the center of mass (CM) between them, it is often more convenient to consider the orbital motion of the smaller object with respect to the larger of the two.
A circular orbit requires a specific tangential velocity at the given separation, with no initial radial velocity to affect the orbit. This situation can occur when two celestial bodies pass near each other at the correct velocities and separation, thus resulting in one orbiting the other. Of more immediate interest is the case of sending up a rocket and aiming it so that its tangential velocity results in a circular orbit.
If the tangential velocity is less than for a circular orbit, the object may go into a small elliptical orbit or even fall into the larger object. If the tangential velocity is greater than for a circular orbit, the object may go into a large elliptical orbit or fly off into space in a parabolic or hyperbolic path.
The equations for a circular orbit and a parabolic path are the starting points in considering the effect of velocity. Also, if the mass of one object is much greater than the other, the equations can be simplified.
Questions you may have include:
- What is the starting point for orbital equations?
- What are the orbital paths of the object?
- What conditions would cause the objects to fly off into space?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Starting points for orbital equations
The path the object follows depends on its tangential velocity. The velocity equations for a circular orbit and for a parabolic path are starting points for examining the relationship between velocity and orbital motion.
Circular orbit
The tangential velocity equation for a circular orbit with respect to the other object is:
vT = √[G(M + m)/R]
where
- vT is the tangential velocity in km/s
- M and m are the masses of the two objects in kg
- G is the Universal Gravitational Constant = 6.674*10−20 km3/kg-s2
- R is the total separation between the centers of the two objects in km
If one mass is much greater than the other (M >> m), the equation reduces to:
vT = √(GM/R)
Parabolic path
The equation for a parabolic path is:
vT = √[2G(M + m)/R]
Likewise, if M >> m, the equation becomes:
vT = √(2GM/R)
Orbital paths of object
The possible orbital paths of an object are:
- Small ellipse
- Circular orbit
- Large ellipse
Small elliptical path
When the tangential velocity of an object is less than required for a circular orbit, the object will follow a small elliptical path.
0 < vT < √[G(M + m)/R]
If vT = 0, only the gravitational force along the radial axis would affect the falling object, such that the objects would collide. If vT was slightly greater than zero and both objects were point-masses, the falling object would travel in an elliptical orbit, just missing the other object.
However, if the objects have substantial size, it is possible that the orbiting object will collide with the fixed object. An example of this can be seen with Newton's Cannon, where the cannonball is not propelled fast enough to go into orbit.
(See Gravity and Newton's Cannon for more information.)
Cannonball does not have enough velocity to go into orbit
Also, since the mass of the cannonball is so small compared to the mass of the Earth, the approximation can be made:
0 < vT < √(GM/R)
Circular orbit
At a specific tangential velocity for the given masses and separation, one object will go in a circular orbit with respect to the other object according to the equation:
vT = √[G(M + m)/R]
If M >> m:
vT = √(GM/R)
As noted before, we assume there is no initial radial velocity that would complicate things and affect the shape of the orbit.
Large elliptical orbit
If the velocity of the object is greater than required for a circular object but is less than that of a parabolic path, the object will travel in a large elliptical orbit around the other object:
√[G(M + m)/R] < vT < √[2G(M + m)/R]
If M >> m:
√(GM/R) < vT < √(2GM/R)
Go out into space
If the velocity is great enough, the object will escape the gravitational force and fly out into space. It will travel in a parabolic or hyperbolic path, depending on the initial tangential velocity.
Parabolic path
If the tangential velocity is such that:
vT = √[2G(M + m)/R]
the object will follow a parabolic path and go off into space. This velocity is called the gravitational escape velocity.
(See Overview of Gravitational Escape Velocity for more information.)
Likewise, if M >> m:
vT = √(2GM/R)
Rocket follows parabolic path
Hyperbolic path
If the tangential velocity is is greater than required for a parabolic path, the object will follow a hyperbolic path and go off into space:
vT > √[2G(M + m)/R]
If M >> m:
vT > √(2GM/R)
In this situation, the velocity exceeds the escape velocity.
Summary
The tangential velocity of one object with respect to another determines whether the object collides with the other object, goes into orbit or flies off into space. The velocity equations for circular orbits and parabolic paths are the starting points for establishing the effect of tangential velocity on orbital motion.
The velocity equations are:
Collide or small elliptical orbit: 0 < vT < √[G(M + m)/R]
Circular orbit: vT = √[G(M + m)/R]
Large elliptical orbit: √[G(M + m)/R] < vT < √[2G(M + m)/R]
Parabolic path: vT = √[2G(M + m)/R]
Hyperbolic path: vT > √[2G(M + m)/R]
If the mass of one object is much greater than the other, the equations are simplified.
Be observant and curious about things
Resources and references
Websites
Circular Orbits - Wikipedia
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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