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Astronomy/orbital speed, etc.

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Expert: - 7/14/2011


Question
Dear Mr. Stahl,

Can you answer a few basic questions for me? My brother and I are having a debate about orbital speed in relation to altitude. Assuming for a moment that the orbit is circular or close to it, would not the orbital speed of an object or vehicle decrease with its distance from the earth? For example, would not the Shuttle at at altitude of, say, 250 miles, orbit the globe faster than the space station, which I believe is at a much higher altitude, or a satellite in geosynchronous orbit. That is, the higher the orbit, the slower the speed? Or do I have this backwards? I thought this related to the idea, theorem, etc., that a point closer to the center of rotation (ex: O.D. of a wheel rim) moved faster than a point farther out (ex; O.D. of tire).
I'm willing to take in as much as you feel like commenting on although it's been years since I took calculus. Thanks very much for your time."

Answer
Illustration of Kepler\'s 2nd law
Illustration of Kepler
Hello,

Okay let me give a comprehensive answer, first from a general (elliptical orbit) point of biew following Kepler's 2nd law (see accompanying diagram) and then more specifically to the issue of Earth satellites.

As shown in the diagram, Kepler's 2nd law of planetary motion states equal areas will be swept out in equal intervals of time as a planet makes its elliptical orbit. I've shown two extremum areas, A1 and A2, the first near the perihelion point (or perigee) of the orbit and the second near aphelion or apogee. It can be seen that given equal areas, A1 = A2, then the velocity (specifically the polar component theta of velocity- referred to polar coordinates)at perihelion will exceed that at aphelion since a greater arc (conforming to delta Θ) is swept out. Hence the orbital speed is greater at closer proximity to the Sun, say.

Since you are familiar with calculus, this can also be worked out quantitatively in terms of instantaneous sector velocities. For example the area of the sector A2 bounded by two radius vectors, would be:

delta (A2) = ½ r^2 (delta Θ/ delta t)

Now, in the limit of delta t->0

dA2/ dt = ½ r^2 (dΘ/dt)

Since the components may be expressed:

v(r) = dr/dΘ,   and v(Θ) = r dΘ/dt

then, the given sector (A2) velocity can be expressed:

dA2/ dt = ½  r v(Θ)

and:

v(Θ) = 2/r [dA2/dt}

Clearly, since r1 < r2 (at perihelion) then the orbital velocity must be greater there than at aphelion. Hence, again, the velocity diminishes with the distance or in this case the magnitude of the radius vector.

In the case of Earth satellites, it is often better to employ the Newtonian version of Kepler's 3rd of harmonic law.

Recall that this basically stated:

P^2 ~ a^3

that is, the period squared is proportional to the semi-major axis cubed.

An equality can be arrived at by the inclusion of a constant, k:

P^2 = k a^3

By Newton's law of gravitation, the force of attraction of the Earth for an orbiting object, e.g. satellite, is equal to the centripetal force.

Let R be the Earth's radius, and then:

GMm/R^2 = mv^2/R

Here m is the mass of the satellite. Since m is common to both sides, we have:

GM/ R^2 = v^2/R

Since we are going to need the period P, it is useful to express v, the velocity in terms of period, so:

v = 2π/P

whence:

GM/R^2 = (2π/P)^2 1/R

or, in terms of P^2:

P^2 = (4π^2/ GM) R^3

which is just the Newtonian statement of the Harmonic law.

This needs to be refined, however, for satellite orbits, so we let R = the radius of the Earth, and h the altitude of the satellite above the surface of Earth. Then we may re-express the relationship:

P^2 = (4π^2/ GM) (R + h)^3

It can easily be seen from the preceding, that the larger the value of h or altitude, the greater will be the period, P. The longer P, the slower the velocity.

Hence, the greater the altitude of a satellite, the slower it moves in its orbit. This, of course, is consistent with what we already found in the general case of planetary orbits using Kepler's 2nd law.

Hopefully, this clears up the issue!  

Philip Stahl

Expertise

I have forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics as applied to astronomical investigations.

Experience

Astronomy: more than forty years experience starting with construction of my own simple telescopes. Worked at university observatory in college, doing astrographic measurements. M.Phil. degree in Physics/Solar Physics and more than ten years as researcher.

Organizations
American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union

Publications
Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'.

Education/Credentials
B.A. Astronomy, M. Phil. Physics

Awards and Honors
American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research

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