Gravitational Slingshots

I always wondered why doesn’t the sun slow space probes down when they are leaving the Earth for outer planets. Isn’t there a risk that the probe might change its trajectory and fall into the sun? There is. You see, the more distant the space probe gets from the Sun, the more potential energy it gains. However, energy must be conserved at all costs. Therefore the probe loses its Kinetic energy (and therefore its speed) in order to get away from the sun. It is the same as when you throw a rock up into the air.

But there comes a point, as with the rock, when the probe loses all of its kinetic energy. At that time it has reached as far away from the sun as it can. Yes, you could add thrusters to make sure the probe continues its journey. But the extra weight and inefficiency of propellants known to us make it an unsuitable alternative.

Enter the Gravitational Slingshot! Nature’s way of compensating us (very marginally) for all the millions of years we’ve been dragged through the mud in the name of evolution. Through this method, space probes go into a partial orbit around a planet and emerge on the other side with a greater velocity. “No!”, some might say, because it is a violation of conservation of energy. Intuitively it seems that way, but it is all a matter of relativity.

slingshot

Imagine there is a probe approaching a planet with a velocity ‘u’. To an observer on the planet, the apparent velocity of the probe’s approach will be ‘V+u’, where ‘V’ is the planet’s and ‘u’ is the probe’s heliocentric velocity, i.e. velocity relative to the Sun. It will go into orbit at that speed. Now, when it comes out of orbit on the other side, it is still moving with a velocity ‘V+u’ relative to the planet’s surface. But the planet is also moving in the same direction at velocity ‘V’. So the final velocity as the probe leaves orbit will be ‘V+(V+u)’. Of course, some of that velocity will be reduced due to the planet’s potential, but in the end it will still be greater than the probe’s initial velocity.

If you look at what happened overall, ignoring how it happened, the probe approaches a moving planet at a certain velocity and “bounces off” at a higher velocity. It is just like when you throw a ball at the face of a moving train, the ball bounces off at a higher velocity. Now, the ball changes its momentum (first going in one direction, then another) and transfers that change to the train to ensure conservation. But the train is comparatively so massive that we do not notice the minuscule change in its velocity. That’s the same with planets and probes.

The effective increment in the probe’s velocity is due to the orbited body’s velocity relative to the Sun (analogously, the change in velocity of the rebounding ball depends on the train’s relative velocity to the ground). Of course, the Sun’s velocity relative to itself is zero. Therefore ‘V’ will be zero. So there will be no gravitational slingshot from the Sun (towards planets in its orbit) even though it is the most massive body in the solar system; just like there will be no increment in the velocity of the ball when you throw it at the ground.

Relativistic Doodles

This post explains how my illustration of general relativistic time dilation works. This is the parent post.

Here is an image of a simple classical timeline:

timeline_1

There are 2 observers: green and red. They have their own watches. The red observer shoots an arrow towards the green observer. The position of the arrow vs. the time recorded by the two observers looks like the picture above. If we take a trace of the trajectory of the arrow, we get:

timeline_2

If we were to take the trace of a 2 signals 1 second apart that travel at constant speed, and 2 signals 1 second apart and traveling instantaneously, we would get:

timeline_3

You will notice that the components of all 4 traces (like the ones drawn in grey) are parallel and perpendicular to the space-time axes. This is always the case. So even when I distort the edge of the red timeline a bit, I get:

timeline_4

So if I am standing with the red observer, and I see him sending out signals, I will not notice a difference even when the red timeline distorts, because to me, the components of the signals are still parallel and perpendicular to my space and time. In the same way, when an observer is falling into a gravity well (like that of a black hole) and sends out signals, the observer does not notice the relativistic effects of gravity on the signal. However, the observer who is standing far away from such distortions notices an altered signal. So even though the signal at t=8 was instantaneous according to the red observer, it reached the green observer at t=9.

If you understood this, then you will know what is wrong with this set of traces of signals that a red observer sends as she falls into a black hole:

time_dilation_gravity_WRONG