For example, lets say NASA wants to send a satelite aroud a moon,

if they want this spacecraft to be in a certain orbit, they need to know at what speed it needs to be moving; how are they going to calculate that speed?

by using Newton’s equations or by using some special relativity equations?

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By knowing the mass of the moon, or any celestial body, and the distance a satellite needs to be placed on from the center of the force exerted by the body, gravity. It is quite easy to calculate the required speed to maintain an orbit.

Basically, you need a speed perpendicular to the pull that cancels out the gravity force. This varies with distance. Closer to the center of mass, a lower orbit, means more speed is needed as the gravity pull will be higher.

There is no need to involve relativity in the calculations for a basic orbit around a body. However, satellites need to keep adjusting their orbit due to minimal variations over time that can add up.

They use very detailed simulations based on Newton’s equations. These simulations employ mathematical models of every element of the environment – planetary bodies, gravity, atmospheres, the sun/solar wind. They then define a set of target conditions (basically, where they want to end up), constraints (like how long they want to take to get there, limits on solar exposure, etc.), and controls (the initial velocity, propulsion capabilities, etc.). Since this problem is way too complicated to solve directly, these simulations use numerical methods to come up with a solution. This basically means that they enter guesses for the controls and see where things end up, then tweak the guesses over and over again to home in on the solution. Depending on how complicated the problem is, this can take a fair bit of time and computing power.

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Using nth body simulations, you can find if initial velocity in x,y,z will result in a stable orbit rather than fly off into space or crash into earth. A simple program will iterate through time, say by 0.01 seconds per step, the smaller the step, the more accurate the simulation.

From the Gravitational constant*Mass(Earth) * Mass(sat)/Distance^2 = force of gravity you can write it as dV = (dT * G * Mearth)/D^2 by changing dT each step you can update the corresponding dV of each axis add it to the old velocity then update its position.

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