The problem is the following:

  • let's say we have a miniaturized satellite*, X;

  • we know its position, velocity, acceleration, the forces applied at any given moment;

  • we also know its mass, its dimensions etc;

  • the satellite tries to approach a point in space, let's say A(x, y, z).

    The question is: what forces (xf, yf, zf) should we apply to our satellite at any moment, such that the satellite reaches and stops at point A.

(*) A satellite is a SPHERES (https://en.wikipedia.org/wiki/SPHERES). We don't need to worry about gravity. The satellites move by ejecting CO2.

  • $\begingroup$ This is an important question. Mathematical methods and software tools were developed to approximate the answers for it. But it is in this present form greatly out of scope here. If you could narrow it down to a specific question (like satellite is in equatorial orbit in LEO, how much dv is needed to reach GEO), you might get an answer. $\endgroup$ – b.Lorenz Dec 8 '18 at 9:37
  • $\begingroup$ By the way, Welcome to Wordbuilding SE! $\endgroup$ – b.Lorenz Dec 8 '18 at 9:38
  • $\begingroup$ And a little remark: satellites rarely stop at points. If they do, they fall to the planet they are orbiting. $\endgroup$ – b.Lorenz Dec 8 '18 at 9:39
  • $\begingroup$ You might get a better answer over at space.stackexchange.com or physics.stackexchange.com $\endgroup$ – dot_Sp0T Dec 8 '18 at 9:48
  • $\begingroup$ 1. Thank you! 2. I'm talking about miniaturized satellites, like SPHERES, on ISS, so I don't have to worry about the Earth gravity. $\endgroup$ – satellites Dec 8 '18 at 9:53

In the absence of gravity and atmospheric friction, objects behave according to Newton's Laws of Motion.

  1. An object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
  2. The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma.

  3. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

So if you have a body flying through frictionless, gravity-free space and want to rendezvous with a different body in that space, you do the following.

First you determine the relative velocity vector between you and the other object. It doesn't matter if the other object is stationary or moving with constant velocity, because the question whether an object is moving or at rest is just a question of your frame of reference. All you need to know is how you move in relation to the destination.

Then you determine what velocity vector you want to have. The direction is obvious: towards that object. The speed depends on how long you want the journey to take and how much fuel you want to spend.

Subtract the relative velocity vector from the desired velocity vector and you have the course correction vector you want to perform.

Now how long do you need to thrust in the direction of that correction vector to achieve that velocity correction? This is where the formula $ force = mass * acceleration $ comes into play. Velocity is acceleration by time. So if you know the force of your thruster in Newton, the mass of your vehicle and you want to know how long you need to burn it to reach your desired velocity, the formula is

$ burnTime = \frac { desiredSpeed }{ mass * thrust } $

What, you misplaced the data sheet of your thrusters and don't know how much force they generate? Don't worry, you can calculate that too. Thrusters generate force by applying Newton's 3rd law. When they expel propellant in one direction, they generate a force in the opposite direction equal to the force they put into the propellant. So you just need to measure how much CO2 the thruster consumes (in kg/s) and multiply it by the velocity the CO2 has (in m/s) when it exits the thruster. The result is its thrusting power in Newton.

So, assuming you performed your course correction burn correctly, you are now flying straight towards the destination with your desired velocity.

Now you just need to stop when you are there. This is done by changing your relative velocity to zero with another burn. This burn is performed the same way as the previous one using the same formula. You take your relative speed and burn in the opposite direction until your relative velocity is zero.

All this of course only applies if you are not in the gravity well of a planet. When you are in an orbit, things get far more interesting (and complicated). If you want to gain an intuitive understanding of orbital mechanics, then I recommend playing Kerbal Space Program. Yes, I am serious. After you shot a little green man to the Mun you will have a better understanding of orbital mechanics than any physics lecture could ever give you. If you don't want to spend the $40 (you should!) then there is an old free demo version around. It is very outdated compared to the main game, but all the orbital mechanics implementations are already there.

  • $\begingroup$ Thanks, sadly I can't upvote your comment yet. $\endgroup$ – satellites Dec 8 '18 at 10:37
  • $\begingroup$ Objects always move in accordance with Newton's (or, at higher velocities, Einstein's) laws of motion, even in the presence of gravity and "atmospheric friction", whatever that may be. $\endgroup$ – AlexP Dec 8 '18 at 11:04
  • 1
    $\begingroup$ @AlexP Of course they do, but it isn't that obvious because it isn't obvious where all the forces come from and how they interact. That's why it took thousands of years until Newton proved Aristotle wrong. $\endgroup$ – Philipp Dec 8 '18 at 11:06

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