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Why do orbital speeds decrease further away from the focus? A simple question, but I want to make sure I am understanding this correctly: Is it ONLY a function of the gravity well? As in, the gravitational field is weaker as you move away from the massive body, so the speed decreases? What if the gravitational field was constant through space? Would the orbit's speed then be constant?

This should be a home-run for someone.

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  • $\begingroup$ As Frederic Brünner and I understood the question differently, could you clarify it: Do you want to know why an object's velocity decreases when its distance to the sun increases, or why planets farther away from the sun move slower? $\endgroup$ – Aecturus Nov 22 '13 at 16:51
  • $\begingroup$ Both. For any orbital system. Also, I know it is impossible in this universe, but theoretically, could there exist a gravitational field such that the orbital speed of any object would be constant, regardless of distance from the massive body? $\endgroup$ – Stu Nov 22 '13 at 16:53
  • $\begingroup$ Well, those are two different effects. The first effect, described in Frederic's answer is due to the conservation of angular momentum and the total energy. The second, described in my answer, is due to the condition for a stable orbit. $\endgroup$ – Aecturus Nov 22 '13 at 16:56
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"Focus" is an inconvenient word if you're thinking of changing the potential, because if you do then the orbits are no longer conics and the word kind of loses its meaning. That aside, let me see if I understood your question correctly:

Given a gravitational potential that's spherically symmetric around a central point $\mathbf{r}_0$, and which has a gravitational potential $V(|\mathbf{r}-\mathbf{r}_0|)$, what's the fundamental reason that orbital speeds decrease as $|\mathbf{r}-\mathbf{r}_0|$ increases? Is this due to the gravitational field being weaker at longer distances?

In that case, the answer is that orbital speeds decrease because $V$ itself increases at longer distances. This is simply conservation of energy: $$\frac12m\mathbf{v}^2+mV(|\mathbf{r}-\mathbf{r}_0|)=E,$$ and if $V$ becomes less negative then $v^2$ must be smaller. Thus, potentials where this doesn't happen must have regions where the potential is repulsive from the origin. One such example is $$V(r)=-\frac1r -r,$$ though of course there's no physical system with that behaviour.

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The velocity of an orbit around some central object can be easily calculated for a circular orbit. Let us assume that there is some central Force $F=c\cdot r^\alpha$, where $c$ and $\alpha$ are some constants (for gravity $c=Gm_1m_2$ and $\alpha=-2$).

For a stable orbit, this central force must be equal to the necessary centripetal force (not balance the centrifugal force, which actually does not exist). This centripetal force is given by $F_{\rm Z}=\frac{mv^2}{r}$.

Now, by simply combining the two forces and solving for $v$ we obtain $$v=\sqrt{\tfrac c m\,r^{\alpha+1}}\,.$$

We see that for $\alpha<-1$, the orbital velocity decreases with distance. For a constant force, the orbital velocity actually increases. The velocity would stay constant for a force proportional to $r^{-1}$.

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The orbital speed will increase as the orbital speed decreases as for the fact that the gravitational potential energy will decrease as the orbital radius decreases. Therefore the gravitational field does work on the mass to increase its speed. It would be a violation of the conservation of energy if the orbiting masses speed did not increase. Change in GPE= change KE

Essentially all orbiting objects are falling towards the main gravitationaly dominant body, however in a stable orbit the speed remains constant as the orbital radius( assuming a circular orbit and a symmetrical gravitational field) is constant so no work is done. When the orbiting masses orbital radius decreases, it continues to fall towards the gravitationaly dominant body,however the orbiting mass is getting closer to the planet so that is has now gained velocity. It is the same mechanism which a ball, that initially was stationary will gain velocity as it falls as work is done on it. This is also why comets on highly elliptical orbits gain a significant orbital speed at the perihelion (closest distance to the sun)

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