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I'm having trouble understanding relative angular/tangential speeds at increasing altitudes above Earth's surface. In particular, I find this comparison of tangential velocities on Wikipedia very confusing. According to it, the tangential speed of Earth's surface (465.1 m/s) is different from the tangential speed required to "orbit" at Earth's surface (7.9 km/s). Why are these different values? My understanding of the Earth was always that material on and within the Earth is orbiting the center of the Earth just like satellties do. Time for multiple questions in one post...

  1. Is material on Earth's surface not in free fall around Earth's center?
  2. How are geostationary orbits even a thing? Seems like the only orbit that could be geostationary would be standing on the Earth's surface.
  3. What changes as you orbit further above the Earth's surface? Does your angular speed increase or decrease? Does your tangential speed increase or decrease?
  4. Is magma near the center of the Earth not rotating faster than material in the crust, like in an accretion disk?
  5. Can two objects be orbiting (circularly) at the same altitude but with different tangential speeds?

Thanks in advance!

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    $\begingroup$ Consider what happens if you're standing at the North or South poles, and how you move relative to the centre of the Earth. $\endgroup$
    – user24157
    Nov 18, 2019 at 7:45
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    $\begingroup$ If we were in orbit, we'd be floating. $\endgroup$
    – PM 2Ring
    Nov 18, 2019 at 8:24
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    $\begingroup$ Consider that you can have celestial bodies that rotate pretty fast (Jupiter 9 hrs 56 minutes, Saturn = 10 hrs, 42 minutes) or pretty slowly (Venus 116 days, 18 hrs, Earth's moon 27 days, 8 hrs), irrespective of the speed needed to orbit them - which is a function of mass and distance. $\endgroup$
    – jamesqf
    Nov 18, 2019 at 17:43
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    $\begingroup$ As to #1, free fall is kinda the opposite of "being on Earth's surface". The former means not experiencing the forces associated with gravity, such as the force of your seat against your chair or your feet against the ground while standing. Even sky diving is not really "free fall" once air resistance builds up. $\endgroup$
    – Jeff Y
    Nov 18, 2019 at 19:08
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    $\begingroup$ ::blink:: I'd be interested in hearing how you came by the understanding that "material on and within the Earth is orbiting the center of the Earth". No reputable source should say anything like that which leaves me wondering about plausible-sounding chains of conjecture. $\endgroup$
    – dmckee --- ex-moderator kitten
    Nov 19, 2019 at 0:44

4 Answers 4

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1. Is material on Earth's surface not in free fall around Earth's center?

No. Material on the Earth's surface -- or inside it -- is not in orbit, and so is not in free fall. You can temporarily put yourself into an orbit (and thus into free fall) by jumping up into the air, or jumping off a higher surface. When you do this, you are briefly in a very eccentric orbit (one which would take you very close to the center of the Earth, if the Earth wasn't a solid body) -- but then you hit the ground and are no longer in orbit.

The Earth rotates in the same way that a spinning top rotates; this has nothing to do with orbits.

2. How are geostationary orbits even a thing? Seems like the only orbit that could be geostationary would be standing on the Earth's surface.

Again, the surface of the Earth is not orbiting. The Earth rotates as a rigid body, with (as AtmosphericPrisonEscape noted) residual angular momentum left over from its formation, like a spinning top.

Because your angular speed in an orbit decreases the further away you are from the Earth, there will be a point where it happens to match the Earth's spin rate. If you arrange the orbit so it is above the equator and in the same direction as the Earth's spin, then you will always be above the point on the equator: a geostationary orbit.

3. What changes as you orbit further above the Earth's surface? Does your angular speed increase or decrease? Does your tangential speed increase or decrease?

Both your angular speed and your tangential speed decrease the further away you get. (Your angular speed would decrease even if your tangential speed stayed the same, because the circumference of your orbit increases with altitude; but in fact the tangential speed decreases as well.)

4. Is magma near the center of the Earth not rotating faster than material in the crust, like in an accretion disk?

The Earth rotates approximately as a rigid body, so in general, no. The molten outer core (which is not magma) may rotate slightly slower, while the solid inner core might rotate a little faster, but we're talking about $\sim 0.1$ degrees per year differences, and this has nothing to do with orbits. (The Earth is nothing like an accretion disk.)

5. Can two objects be orbiting (circularly) at the same altitude but with different tangential speeds?

Ignoring minor deviations due to things like the non-spherical nature of the Earth, mass concentrations in the crust, etc., the orbital speed for a circular orbit is a function of the altitude only. So two objects in circular orbit at the same altitude must have the same tangential speed. (Note that they can have different velocities, because velocity is a vector quantity -- so you can have two object orbiting in different -- even opposite -- directions at the same altitude, at least until they run into each other.)

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    $\begingroup$ Nice answer, but minor correction from geophysics: the Earth's Outer Core is molten (mostly made out of iron) and convecting according to magnetohydrodynamics, maybe(!) rotating slightly more slowly on average. The Earth's Inner Core consists (mostly) of solid iron and may be rotating slightly faster then the mantle/surface (but no more than 0.1 deg/year or so faster - with the years this estimated number went down substantially) $\endgroup$
    – frederik
    Nov 18, 2019 at 15:48
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    $\begingroup$ "You can temporarily put yourself into an orbit (and thus into free fall) by jumping up into the air, or jumping off a higher surface...but then you hit the ground and are no longer in orbit." To paraphrase Douglas Adams, "The knack [to flying] lies in learning how to throw yourself at the ground and miss." $\endgroup$
    – BruceWayne
    Nov 18, 2019 at 17:15
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    $\begingroup$ @JeffY You're using a more restricted definition of an orbit. In general, anything that follows geodesics is an orbit (or in Netwonian models, any trajectory curved mostly by gravity), and we usually allow for orbits that don't perfectly follow geodesics too (e.g. satellites around Earth do not follow geodesics perfectly, because they're accelerated by their collisions with air). In the same sense, a person jumping off a cliff is imperfectly following a geodesic (until he hits the ground). The main point is that objects in orbits move exactly the same way regardless of how many orbits you do $\endgroup$
    – Luaan
    Nov 19, 2019 at 9:18
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    $\begingroup$ @JeffY Even in the two-body problem, there are hyperbolic orbits (like extrasolar comets!) which are not "cyclic". And more generally, there are chaotic orbits which are not cyclic, either. (As Luaan noted: geodesics!) $\endgroup$
    – Peter Erwin
    Nov 19, 2019 at 18:37
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    $\begingroup$ "5. Can two objects be orbiting (circularly) at the same altitude but with different tangential speeds?" Can't let that go without mentioning horseshoe orbits $\endgroup$
    – candied_orange
    Nov 20, 2019 at 17:03
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Imagine you are in orbit around the earth, several 100 km upwards. What happens when you slow down? That's right, you fall down until some force stops your fall. That force is the pushback from the ground.

So next imagine: What happens when you throw a ball in the air? It falls back down to the ground. So it follows, that the ball is too slow to be in orbit.

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An interesting corollary to this question: if the ground is not in orbit, how does it move (roughly) in a circle? If we model a section of ground as an isolated particle, it's clear that in order to move in a circle despite having a relatively low tangential velocity, it would need to have an ongoing force being applied to counteract the direction the particle would 'like to go', following gravity.

Where does this force comes from? It comes from the electromagnetic repulsion of the nearby material making up the rest of the Earth, which, having been around for a fairly long time, has largely stabilized to an equilibrium where the built-up compression counteracts the force of gravity, allowing material on the surface to move roughly in a circle, despite moving too slowly to be in a free-falling circular orbit.

Normally we think of the ground as 'stopping our fall', which it does, but it also continuously pushes us to keep following the path of the surface's rotation while we're in contact with it. Fundamentally, electromagnetism is winning out over gravity, preventing our collapse and allowing us to move in a rotating frame without needing to orbit.

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  • $\begingroup$ "it also continuously pushes us to follow the path of the surface's rotation while we're in contact with it" - this isn't really true. The material we were made from, and that our parents were made from, and that life itself was made from, was spinning long before we were around. We just have the momentum too - that's why jumping doesn't catapult you west. $\endgroup$
    – scatter
    Nov 19, 2019 at 17:43
  • $\begingroup$ Well, the residual momentum explains the part that has us moving in a rotating frame in the first place; the force from the Earth explains why that doesn't collapse due to gravity, despite that rotating frame not moving us fast enough to be in a free fall orbit $\endgroup$
    – Dan Bryant
    Nov 19, 2019 at 17:46
  • $\begingroup$ good point, though; I edited the wording to be a bit more precise $\endgroup$
    – Dan Bryant
    Nov 19, 2019 at 18:03
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I find this comparison of tangential velocities on Wikipedia very confusing.

…

According to it, the tangential speed of Earth's surface (465.1 m/s) is different from the tangential speed required to "orbit" at Earth's surface (7.9 km/s).

That might be, but they have explicitly elaborated "… Earth's own rotation at surface (for comparison— not an orbit) …"

enter image description here

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