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I'm intrigued by the idea of launching from the top of a space elevator perched on the top of a Ecuadorian mountain, launching Eastwards to take advantage of Earth's spin.

I'd like to just understand a bit more of the maths to do with fuel consumption, payload, getting to the Moon, etc.

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    $\begingroup$ Have a look at the patched conics method on wikipedia. It still requires the use of the two body equations but you should be able to come up with some rough trajectories using a calculator and paper. $\endgroup$ – PeteBlackerThe3rd Jul 17 at 9:01
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    $\begingroup$ There are some related questions here that can provide a starting point. e.g. space.stackexchange.com/questions/34504/… $\endgroup$ – Hobbes Jul 17 at 11:20
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Here is a list of equations you'll need to determine orbital velocity at different points, orbital velocity required to orbit, apoapsis, periapsis, mean anomaly, true anomaly, and eccentricity. This is the best I can give you as there is no formula that calculates fuel consumption because it entirely depends on the type of engine and etc.

To determine the orbital speed at different points in an elliptical orbit, you'll have to use the Vis-Viva equation. $$v=\sqrt {\mu (\frac{2}{r}-\frac{1}{a})}$$ where $\mu$ is the standard gravitational paramer (GM), $r$ is the distance between the centres of 2 objects, $a$ is the semi-major axis of the orbit, and $v$ is the velocity.

To determine velocity required to orbit, you need the Orbital Velocity Formula which is $$v = \sqrt{\cfrac{\mu}{r}}$$ Again the variables mean the same as the Vis-Viva equation. And a side note, this equation is for a circular orbit.

To determine the apoapsis and periapsis of an orbit (which also determines the shape of your orbit (eccentricity), see here), you have to find the specific energy, the semi-major axis, and the eccentricity vector. This answer is the best at explaining it.

The mean anomaly is the fraction of an elliptical orbit's period that has elapsed since passing the periapsis. It is measured in an angle which can be used to calculate where a body is in its orbit. The vertex of the mean anomaly is the center of the ellipse The equation to determine the mean anomaly is Kepler's Equation. It can also be solved by $M=n(t-t_0)$, where $t$ is time at arbitrary time (any time you choose) and $n$ is mean motion which is calculate by $n=\frac{2\pi}{P}$, where $P$ is orbital period (how long it takes to orbit).

The true anomaly again, similar to the mean anomaly, defines where a body is in its orbit. Instead, the vertex of the true anomaly is the focus of the ellipse (the planet or central body). Here is how to calculate the true anomaly.

To calculate the delta v from a gravity assist, simply use the equation found in this source. It is a multi-step calculation and this site can describe it better than I can. The gravitational assist formula can also be found in this Stack Exchange question.

These are all the things necessary to calculate an orbit or to find a body in an orbit. There might be more so if you think i'm missing something, let me know or feel free to edit.

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