# Orbital speed for an inclined circular orbit round a spinning spherical planet

For the purpose of this question, I'm assuming that

• The Earth is perfectly spherical
• The Earth is made of material whose density, if it varies at all, depends only on the distance from the center of the Earth
• The Earth spins, completing a rotation once every 24 hours.
• The mass of the Earth is much greater than that of the satellite my question is about
• The Earth is far away from the gravitational influence of any other celestial body.

Also assuming the absence of any atmosphere, or solar wind, or any solar radiation in the vicinity.

If we have a satellite orbiting the Earth in an circular orbit that is inclined at, say, 45° to the Earth's equatorial plane, is there any factor (however small) in Newtonian mechanics that can cause the speed (in meters per second) of the satellite at any instant to vary with its distance from the equatorial plane (and thus the latitude of the sub-satellite point) at that instant?

Or, in summary, does the spin of a planet affect the orbital speed of a satellite going around it according to Newtonian mechanics?

I ask this because I saw a web-page somewhere (which I can't find now, unfortunately) that stated that this speed is different between when the satellite is over high latitudes and when it is crossing the equator, and that this is due to conservation of angular momentum. Wanted to understand whether this is legit, and if so, how.

Edit: Ok, so the web-page I read is this one and I may have misread it. What it does actually say, unfortunately under the heading 'Inclined Orbits', is that for a satellite in circular orbit that isn't actively station-keeping

for reasons beyond the scope of this article (conservation of angular momentum), the orbit assumes a slightly elliptical shape

I'd like to understand the reason for that instead.

• The Earth is perfectly spherical
• The Earth is made of material whose density, if it varies at all, depends only on the distance from the center of the Earth
• The Earth is far away from the gravitational influence of any other celestial body.
• Also assuming the absence of any atmosphere, or solar wind, or any solar radiation in the vicinity.

These are the necessary Hypothesis to make the orbit a Keplerian orbit.

From that we know the trajectory can either be circular, elliptic, or hyperbolic.

• The mass of the Earth is much greater than that of the satellite my question is about
• The Earth spins, completing a rotation once every 24 hours.

All irrelevant details.

For #1, the orbit exists is not around the main body's CoG, but around the barycente of both CG. But is does simplify visualization.

Earth spin does not affect the gravity field.

If we have a satellite orbiting the Earth in an circular orbit [...]

There is your wrong assumption. You assumed a circular orbit. If the orbit is circular, the effect you seek is not observable.

(...) is there any factor (however small) in Newtonian mechanics that can cause the speed (...) of the satellite at any instant to vary with its distance from the equatorial plane

If the orbit is non-circular, yes. The Vis-viva equation tells us:

v² = GM(2/r - 1/a)


Where:

• v is the speed
• r is the radius
• G, M are the gravitational constant and earth mass (also constant)
• a is the semi-major axis, also a constant (for that orbit)

You can see that whenever r varies, so does v. You can also better understand why I said under circular orbit (where by definition r is constant), v must stay constant.

Or, in summary, does the spin of a planet affect the orbital speed of a satellite going around it according to Newtonian mechanics?

That is not the summary, as the spin has no effect on the orbit.

## However

As the satellite orbits, it traces a path on the surface of the earth, also called theground track. As the body rotates underneath it, the path may take many shapes, including simple circle; sinuses, tracing back on itself, butterfly shapes, or a simple dot for geostationary satellites.

Understand that the orbit is completely unchanged by the spin. Only the projection on the moving ground is affected.

These track's shapes will be affected by:

• the ground speed (the spin speed)
• the inclination of the orbit
• and the altitude (which is linked to the speed of the satellite, as explained above) of the statellite if its orbit is eccentric

That's not it, all the other orbital elements will affect the path like for example the argument of the ascending node.

• Thank you Sylvain, 'spin doesn't affect the orbit' is the confirmation I was looking for. – Avijit Jun 15 '17 at 17:27

The motion of a mass in a $1/r^2$ gravitational potential such as the Earth's can be parabolic, hyperbolic, elliptic or circular, depending on the energy of the mass. In other words, there is no specific reason why the orbit of any object around the Earth should be perfectly circular. Ellipses are perfectly good solutions to the equations of motion as well. If this is the case, then the Earth must sit in one of the foci of the ellipse.

Now, due to conservation of angular momentum, at the points of its (elliptic) trajectory where the satellite is closest to the Earth, its speed must be higher to make up for the lost "radius" term in the definition of angular momentum; when it's farther away, it's slightly faster. This is equivalent to Kepler's second law.

As I said, there is no particular reason the orbit is elliptic, other than the fact that we haven't explicitly tried to make it circular.