Circular Planetary Orbits

Although the gravitational orbits of the planets around the Sun are slightly elliptical, they are typcially considered circular for ease of calculations. This is also true for the orbits of moons around the planets, as well as the various satellites in orbit around the Earth.

In order for the path to be circular, the orbiting object must have a certain velocity for the masses involved and the separation between the objects. If the velocity is different, the object may fall due to the gravitational force, go into an elliptical orbit or even fly off into space.

You can find the orbiting velocity of an object by equating its centrifugal force with the gravitational force between the two objects. When the mass of the orbiting object is much smaller than that of the other object, an approximation of the general equation for velocity can be made.

In reality, the two objects are actually orbiting the center of mass between them. Their speeds are such that the separation between them is constant. You can derive the velocity of each object with respect to the center of mass. The orbital velocity of one object with respect to the other is then the sum of the two velocities around the center of mass.

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Requirements for circular orbit

There is a specific linear tangential velocity required for an object to be in a circular orbit at a given separation around a much larger object.

If velocity not great enough

If the tangential velocity is not great enough, the smaller object will fall toward the large object.

One example is that of a low orbit space satellite that has been slowed by air resistance such that it soon falls back to Earth.

If velocity too great

If the velocity is too great for the given separation, the object may take an elliptical orbit or even fly off into space.

Most planets follow slightly elliptical orbits around the Sun, being affected by its orbital velocity and the gravitation from other planets. Mercury has an elliptical orbit that has the greatest eccentricity of the planets, while the Earth’s orbit is almost circular.

Halley’s Comet follows a large elliptical orbit, being visible in the Solar System about every 75 years.

If the velocity of a comet is too great for it to go into orbit around the Sun, it may pass by and then fly away into space again along a parabolic path.

If velocity just right

The velocity of the orbiting object is just right for the given altitude when the centrifugal inertial force caused by its velocity equals the gravitational force at the specific separation from the larger object. In such a case, the smaller object will go into a circular orbit around the larger object.

Deriving the velocity for circular orbit

The circular path of an orbiting object—like a satellite orbiting the Earth—is similar to the situation where you swing an object around you that is tied to a string. The force you apply to the string equals the tendency for the object to fly in a straight line. Likewise, the gravitation from the Earth must equal the inertial force of the moving satellite to put it into a circular orbit.

You can find the velocity required for a small object to be in a circular orbit around a larger object by comparing the gravitational force with the centrifugal inertial force and then solving for the velocity.

Gravitational attraction

From the Universal Gravitational Equation, the force of attraction between objects is:

F = GMm/R²

where

• F is the force of attraction between two objects in newtons (N) or kg-m/s²
• G is the Universal Gravitational Constant (6.674*10⁻¹¹ N-m²/kg²)
• M is the mass of the much larger object in kg
• m is the mass of the smaller object in kg
• R is the separation in meters (m) between the objects, as measured from their centers of mass

Centrifugal inertial force

The inertia on an object in orbit around the other object causes a centrifugal force that pulls it outward. This is because the object tends to continue its movement in a straight line. The centrifugal force is:

F_c = mv_T²/R

where

• F_c is the centrifugal inertial force in N or kg-m/s²
• v_T is the tangential velocity in m/s

Solve for velocity

In order for the object to be in a circular orbit, the centrifugal force must equal the gravitational force:

F_c = F

mv_T²/R = GMm/R²

Solving for v_T:

v_T² = GM/R

v_T = √(GM/R)

Note that the mass of the object in orbit is not a factor in this equation.

Verify units

Verify the equation is correct for the units used:

v_t m/s = √[(G m³/kg-s²)(M kg)/(R m)]

m/s = √[(m³/kg-s²)(kg)/(m)]

m/s = √(m²/s²)

m/s = m/s

Orbiting the center of mass

The actual situation of an object orbiting another is that both objects are orbiting the center of mass (CM) or barycenter between them.

In order to have circular orbits, the velocities of the objects must be such that the separation between the objects remains constant.

Separation between objects

The separation between the orbiting objects is:

R = r_M + r_m

where

• R is the fixed separation between the objects
• r_M is the separation between the center and the CM of the object of mass M
• r_m is the separation between the center and the CM of the object of mass m

The lengths of r_M and r_m are determined from the masses of the objects:

r_M = mR/(M + m)

r_m = MR/(M + m)

(See Gravitation and Center of Mass for more information and derivation.)

Gravitational force

Since the separation R is fixed, the force between to the two objects remains:

F = GMm/R²

Centrifugal force

The centrifugal inertial force on each object relates to its circle of travel:

F_M = Mv_M²/r_M

F_m = mv_m²/r_m

where

• F_M is the centrifugal inertial force on mass M
• v_M is the tangential velocity of mass M
• F_m is the centrifugal inertial force on mass m
• v_m is the tangential velocity of mass m

Substituting for r_M and r_m in the equations:

F_M = Mv_M²(M + m)/MR

F_m = mv_m²(M + m)/MR

Solve for individual velocities

Since the centrifugal force equals the gravitational force for a circular orbit, you can solve for the velocity. In the case of the object with mass m:

F_m = F

mv_m²(M + m)/MR = GMm/R²

v_m²(M + m)/M = GM/R

v_m² = GM²/R(M + m)

v_m = √[GM²/R(M + m)]

Likewise, for the object of mass M:

v_M = √[Gm²/R(M + m)]

Find sum of velocities

Although both objects are moving with respect to the center of mass, they are also moving with respect to each other. The apparent velocity of the one object as seen from the other is the sum of their velocities around the CM.

For example, our view of the Moon is that it is moving with respect to the Earth. However, since the Earth is also moving about the CM between them, the velocity of the Moon, seen from the Earth is the sum of their velocities around the CM. Likewise, the Earth would seem to move at the same velocity, as seen from the Moon.

The sum of the velocities is:

v = v_m + v_M

v = √[GM²/R(M + m)] + √[Gm²/R(M + m)]

Some algebraic manipulation is necessary:

v = M√(G)/√[R(M + m)] + m√(G)/√[R(M + m)]

Combine both fractions over same denominator:

v = [M√(G) + m√(G)]/√[R(M + m)]

v = [(M + m)√(G)]/√[R(M + m)]

Note that (M + m) = √(M + m)²:

v = √[(G)(M + m)²/R(M + m)]

Thus:

v = √[(G)(M + m)/R]

This is the orbital velocity as seen from either object.

You can find the orbiting velocity of an object by equating its centrifugal force with the gravitational force between the two objects. Since the two objects are actually orbiting the center of mass between them, you can derive the velocity of each object with respect to the center of mass. The orbital velocity of one object with respect to the other is then the sum of the two velocities around the center of mass. The resulting equation for orbital velocity is:

v = √[(G)(M + m)/R]

If M is much greater than m, the equation reduces to:

v = √(GM/R)

See the Side Menu for more Gravity and Gravitation topics

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Use possibility thinking

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Resources

The following resources provide information on this subject:

Websites

Orbital Mechanics – Rocket & Space Technology by Robert A. Braeunig

How Satellites Work – HowStuffWorks.com

Orbit – Wikipedia

Circular orbit – Wikipedia

Acceleration due to Gravity Calculations – from Western Washington University

Gravity and Gravitation Resources

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Mini-quiz to check your understanding

If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.

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Also see Answers to Readers’ Questions.