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There was this question we did in class. There is a satellite orbiting the Earth. It runs into some particles, and due to friction, the satellite loses speed. We were asked to predict what will happen now and why ( assuming it doesn't run into a rough patch again and doesn't get to the Earth's atmosphere).

I came up with this:

  1. Speed decreases so the the satellite starts moving closer to Earth.
  2. As it does so, the force of gravity gets stronger and again the object moves closer to the Earth.

Is this right somewhat? The reason I doubt is that the satellite loses kinetic energy due to friction, but my prediction also makes it lose potential energy without making up for it ( Maybe this speeds up the object in the direction in line with the gravitational field sort of like when a ball drops from a height?) Also, my teacher said the satellite should gain tangential velocity, but I could not identify any force acting in this direction so as to speed up the satellite. How did tangential velocity increase?

Basically, any help on how my thinking is off and what I need to consider in this situation will be helpful.

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  • $\begingroup$ What it means by the statement ''It runs into some particles, and due to friction, the satellite loses speed.'' ? $\endgroup$
    – user213933
    Commented Apr 20, 2019 at 16:51
  • $\begingroup$ @Shreyansh, it experiences friction so loses kinetic energy as heat. $\endgroup$
    – Sal_99
    Commented Apr 20, 2019 at 16:58
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    $\begingroup$ See this very similar question: physics.stackexchange.com/q/474175 $\endgroup$
    – Gert
    Commented Apr 21, 2019 at 23:28
  • $\begingroup$ @gert, what I don't get even after looking at the answers there is what exactly happens. I'm getting the sense that it has something to do with the paths becoming elliptical and there being a net force in the tangential direction. But will the satellite crash? $\endgroup$
    – Sal_99
    Commented Apr 23, 2019 at 3:42
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    $\begingroup$ If the friction continues then of course it will crash. what exactly happens would need an equation of motion but appears that that is very very difficult to set up. Hence the worded descriptions of what happens. $\endgroup$
    – Gert
    Commented Apr 23, 2019 at 14:32

3 Answers 3

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You are close. Friction (drag) with gas molecules in the uppermost reaches of the atmosphere causes the satellite to lose orbital velocity. Gravity then pulls it into a lower orbit which means it has to orbit faster to conserve angular momentum while the friction causes it to start warming up. Moving faster means more friction losses, and the satellite falls into a still lower orbit in which it must move even faster and it gets even hotter. Meanwhile, the friction gets worse as the atmosphere becomes denser and as the satellite moves faster, so the satellite gets hotter still. At some point the satellite is no longer orbiting the earth: it is falling faster and faster and it begins to melt from the heat and burn up in the atmosphere.

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  • $\begingroup$ what force is responsible for this increase in velocity? $\endgroup$
    – Sal_99
    Commented Apr 20, 2019 at 17:47
  • $\begingroup$ There is a relationship between orbital velocity and orbital radius. If you reduce velocity, you fall deeper into the gravity well. As you fall, you speed up. the energy required to speed up comes from the difference in gravitational potential energy between a larger radius and a smaller one. $\endgroup$ Commented Apr 20, 2019 at 17:54
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    $\begingroup$ pardon my naive doubt but is it not true that we need a force to act along the tangential direction for the object to speed up? I tried the conversion you suggested and the math made sense but I can't see exactly how the tangential velocity was effected if there was no force in said direction. $\endgroup$
    – Sal_99
    Commented Apr 20, 2019 at 18:05
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    $\begingroup$ I see. One of the good things about answering questions here is it requires you to think about things very thoroughly- and this is a good example. Let me think about this! $\endgroup$ Commented Apr 20, 2019 at 18:56
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    $\begingroup$ The action of friction will make the shape of the orbit to be non-circular, thus the gravitational force will have a component tangential to the velocity (or the velocity will have a non zero component pointing to the centre of the Earth). $\endgroup$ Commented Apr 21, 2019 at 1:57
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Suppose an initial circular orbit, with a tangential velocity $v_0$.

If there is a tangential force againt the movement, the dust drag, that tangential velocity decreases to $v_1$.

If it is only a temporarily dust, the effect is to change the orbit to an eliptical form. The initial angular momentum $|\mathbf L_0| = |\mathbf r||\mathbf v_0|$ decreases to $|\mathbf L_1| = |\mathbf r||\mathbf v_1|$. But now, if there was no more drag, the speed would no more be constant and perpendicular to the radius.

If there is another temporarily dust after some time, the instantaneous tangential velocity decreases again, leading to a further decrease of the angular momentum.

A continuous atmosphere can be understood as a series of several steps of reduction of angular momentum. If the planet was gaseous until the core, the process would terminate with the object stopped in the center ($\mathbf L = 0$). For rocky planets like ours, it ends with a collision with the surface, keeping still some velocity before the impact.

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In a circular motion we have a centripetal force and a tangential velocity. Acceleration due to centripetal force is perpendicular to the tangential velocity. The magnitude of acceleration is such that it changes the direction of velocity so that line of force in the new position of object is perpendicular to the new velocity, pointing towards a fixed point namely center of the circle. This ensures a fixed distance of particle in its motion from the center.
Now when velocity decreases as in the above case but force remains same then the the path of object changes as the centripetal acceleration is same causing it to move towards center is same but the tangential velocity which kept it moving forward decreased. This results in shifting of velocity vector of particle inside the circle which was tangential initially and further increase in force and further deformation of path. No further drag is necessary to pull it down, just for once is enough to destabilize the satellite.

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