Gravitas is a collection of orbital mechanics simualations. Right now, I have n-body gravity and hohmann transfer simulations. I hope to add lagrange points simulation in a future update..

Live Demo: https://gravitas-eta.vercel.app/

I like physics and math and plan to major in either one in the future. I also have college application coming up. This made a great personal project to showcase my passion for physics in my college applications.

The n-body gravity problem asks: given N objects that all gravitationally attract each other at the same time, how do they move?

Unlike the two-body problem (one planet and one star), there is no closed form solution for N ≥ 3 bodies, so you have to simulate it step by step.

This simulation runs the full solar system, in which, every planet exerts a gravitational force on every other planet, in every frame, and the positions are updated accordingly. You can zoom, pan, add new bodies by clicking the canvas and see how the affect the system (collisions, merges, ejected out of system), and switch to a binary star system as well..

A Hohmann Transfer is a two-burn maneuver thats moves a spacecraft between two circular orbits using the minimum possible change in velocity (hence, most efficient way). The first burn is to leave the first orbit and enter an elliptical transfer orbit, and the second burn, when the spacecraft arrives at the second orbit, to circularise.

This simulation calculates the two burns, transfer time, phase angle for any two planetary orbits (in planet mode), and animates the spacecraft travelling the path. Satellite mode does the same for orbits around any solar system body, in km, and allows you to change the altitude and inclination as well.

git clone https://github.com/Hiba-Malkan/gravitas.git

cd gravitas
python3 -m http.server 80800

Then open http://localhost:8080 in your browser.

Or open index.html directly, no need for starting the server.

-- HTML

-- CSS

-- Javascript

$${\overrightarrow{a}}_i=G\sum _{j\ne i}\frac{m_j({\overrightarrow{r}}_j-{\overrightarrow{r}}_i)}{|{\overrightarrow{r}}_j-{\overrightarrow{r}}_i{|}^3}$$

Force between every pair of bodies, all added together. G = 4π² in AU.

$$|\overrightarrow{r}{|}^2\to |\overrightarrow{r}{|}^2+{\epsilon }^2$$

A small ε² term added to the distance to prevent the force from tending to infinity during close encounters (so accelaration doesn't tend to infinity and bodies don't go zooming across the canvas). Used ε = 0.001 AU.

$$k_1=f(t,y)k_2=f!(t+\frac{h}{2},,y+\frac{h}{2}{k}_1)k_3=f!(t+\frac{h}{2},,y+\frac{h}{2}{k}_2)k_4=f(t+h,,y+h{k}_3)$$

$$y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$$

4th order Runge-Kutta integrator evaluates the derivative at 4 points per step and takes a weighed average. This keeps orbits stable long term unlike the Euler method I was previously using.

$$v=\sqrt{GM(\frac{2}{r}-\frac{1}{a})}$$

Vis-viva equation gives the speed of an object at distance r from the central body, in an orbit with a semi-major axis, a. This is used for all Hohmann Transfer velocity calculations. For a circular orbit r = a, so the equation simplifies to v = √(GM/r).

Semi-major axis of the transfer ellipse is:

$$a_{transfer}=\frac{r_1+r_2}{2}$$

Delta-v (change in velocity) at each burn is calcualted using the vis-viva difference between circular velocity and transfer ellipse velocity at each endpoint.

Transfer time, then, (half of the period of the ellipse):

$$t_{transfer}=\pi \sqrt{\frac{a^3}{GM}}$$

Phase angle is the specific angular separation required between a spacecraft and its target planet (or two planets) at the moment of launch to ensure they meet at the destination after a Hohmann transfer orbit (defination taken from Google AI overview).

$$\varphi =180°\cdot (1-\frac{t_{transfer}}{T_{target}})$$

$$M=E-e\sin (E)$$

M is mean anomaly (linear in time), E is eccentric anomaly , e is eccentricity. This has no closed-form solution and hence is solved numerically using Newtons-Raphson iteration. It is used to compute the spacecraft's position on the ellipse at each animation frame.

True anomaly from eccentric anomaly:

$$\tan \frac{\nu }{2}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}$$

Used in the simulation to combine the plane change with the circularisation burn using the law of cosines:

$$\mathrm{\Delta }{v}_{combined}=\sqrt{v_t^2+v_c^2-2v_t{v}_c\cos (\mathrm{\Delta }i)}$$