Planet Simulation Newtons Law - Downscaling

Question

I would like to scale-down the original numbers of our planet's motion, as i cannot properly visualize it in Unity3D (Game-Engine).

I have:

1) Initial Position (-3.5e10, 0) (km)

2) Initial Velocity (0.0, 1.4e03)

3) Mass of Planet 3.0e28

I would like to have the Position in an approximateley range of [-100, 100]. So can i just divide the Position and Velocity just by 1000000000. ? Then my Initial Pos would be 35. As Velocity is as well km / s. is my conversion ok?

Thank you!

Answer 1

Well I'm not too sure with what others said. I think you have to me more careful. If you have mass particle moving in central gravitational field(around the Sun) than it moves according to these equations: \begin{align} \dot x &= v\\ m \dot v &= - G \frac{mM}{\|x\|^2} \end{align} You are looking for scaled trajectory $\alpha \tilde x = x $. Scaled trajectory satisfy this equation: \begin{align} \alpha \dot{ \tilde x} &= \alpha \tilde v\\ m \alpha \dot{ \tilde v} &= - G \frac{mM}{\alpha^2\|\tilde x\|^2} \end{align} This can be changed to \begin{align} \dot{ \tilde x} &= \tilde v\\ m \dot{ \tilde v} &= - G \frac{m \tilde M}{\|\tilde x\|^2} \end{align} where $\tilde M = \frac{M}{\alpha^3}$. So if you scale your initial speed by $2$ than you have to scale your Sun mass by $2^3$.

I will think what has to be done when you have $n$ planets acting on each other.

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edit: It is same for $n$ planets. They move according to these equations

\begin{align} \dot x_i &= v_i\\ m_i \dot v_i &= - G \sum_{j\neq i} \frac{m_i m_j}{\|x_i - x_j\|^2} \end{align} Scaling it with $\alpha \tilde x = x$: \begin{align} \alpha \dot{ \tilde{ x_i}} &= \alpha v_i\\ m_i \alpha \dot{ \tilde{ v_i}} &= - G \sum_{j\neq i} \frac{m_i m_j}{\alpha^2 \|\tilde x_i -\tilde x_j\|^2} \end{align} Change last equation to $$ \tilde m_i \dot{ \tilde{ v_i}} = - G \sum_{j\neq i} \frac{\tilde m_i \tilde m_j}{\|\tilde x_i -\tilde x_j\|^2} $$ where $ \alpha^3 \tilde m_i = m_i$.

Conclusion is the same if you want to scale your simulation with factor $\alpha$ you have to scale position and velocity by $\alpha$ BUT mass of planets has to be scaled with $\alpha^3$.

Answer 2

I would recommend using SI units (meters instead of kilometers) when doing the calculations.

Other than that, yes, if you divide each factor by the same value, the simulation is still accurate.

Answer 3

If you are linearly scaling the space shrinking it $K$ times then you should "shrink" the speed the same way:

$x^{scaled} = \frac{x}{K}, y^{scaled} = \frac{y}{K}, z^{scaled} = \frac{z}{K}$

$v^{scaled} = \frac{v}{K}$.

Note that areas are scaled by $K^2$ and volumes by $K^3$. This implies the change of masses as well, as were mentioned by @tom.