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I understand that when a Earth + Satellite system forms (satellite comes into orbit) the total energy is -ve. If the energy of the system when placed at infinity is 0 (assuming P.E and K.E is 0 at infinity), this means that when the satellite comes into orbit the system has lost some energy. The fall in P.E is not compensated by increase in K.E. So my questions are:

  1. Where has the decrease in total energy gone?

  2. Since total energy is decreased, as per $E=mc^2$ has the mass of the system decreased?

Qmechanic
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Alex
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3 Answers3

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The fall in P.E is not compensated by increase in K.E.

Actually it is, unless the satellite is acted on by some external force. If we just let the satellite free fall from infinity, then it has an increasing velocity and hence increasing kinetic energy as it approaches the Earth. The sum of its kinetic energy and its potential energy at any point $P$ must be constant. So if we take its KE and PE at infinity to be zero then we have

$KE_P + PE_P = KE_{\infty} + PE_{\infty} = 0$

and since $KE_P > 0$ we have $PE_P < 0$.

Of course, we can introduce an external force to act on the satellite and reduce its kinetic energy to zero. For example, we could give the satellite thrusters and fire them so that it is stationary (relative to the Earth) at some point $P$. So we have $KE_P = 0$ and $PE_P < 0$. But now we have to take into account the energy of the propellant fired out by the thrusters, and this balances the potential energy lost by the satellite.

gandalf61
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Where has the decrease in total energy gone?

As gandalf61 has already pointed out, this question is ill-posed. Objects simply do not get into a bound orbit with each other unless they can eject that energy difference from the system. For astronomical bodies, that's usually other gas/particles/smaller bodies which get accelerated by the two bodies you are considering. The gas would get heated up by the stirring effect of the two bodies, and the particles/smaller bodies would usually get ejected from the vicinity. So, that's where the energy difference goes when two objects capture each other into orbit.

Since total energy is decreased, as per E=mc2 has the mass of the system decreased?

Yes. That mass was carried away by the smaller bodies which helped the two bodies get into orbit.

Tangent:

The earth-moon system did not form that way. Instead, the leading theory is that two protoplanets crashed into one another in a cataclysmic event. This collision was not heads-on, but rather with a significant offset. That offset brought a lot of angular momentum into the system, and allowed the debris to settle down into two distinct drops of lava. We know the bigger one as our home planet, and the smaller one as its moon.

After the event, the big lava drop was rotating extremely fast, and obviously prograde with the smaller lava drop's orbit. As such, the tidal forces subsequently slowed down earth's rotational speed to a little under 24h, and lifted the moon's orbit high up to where it is today. This transfer of angular momentum continues even today, moving the moon's orbit further out by about 38mm per year.

cmaster - reinstate monica
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Any bound system (no matter gravitationally or otherwise bound) has less energy and therefore less mass than the constituents separated at infinite distance, measured together.

If you (in theory) start with a planet, a wannabe satellite at an infinite distance and no other bodies handy, getting the body into a closed orbit around the planet is not possible by purely gravitational means. You will get a parabolic or a hyperbolic orbit and the body will go back far away into space.

This has practical implications - e.g. when going to Moon, a space vehicle has to do "braking" or "orbit insertion" burns near the target in order to enter an orbit. Otherwise, it either hits the target or flies away.

When you do the orbit insertion burn, the "extra" energy goes into the propellant that goes out of the engine nozzles.

If the target has atmosphere, one may as well aim at atmosphere's upper layers and use it for braking, sparing some propellant. The details go out of the scope of the question, but in this case the extra energy heats up the atmosphere, as well as the space vehicle itself.

fraxinus
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