0

I am looking for a simple formula x(t) to calculate the position of a planet.

Simple means, the planet is moving on a two dimensional plane following the curve of an ellipse.

My problem is not the curve of an ellipse. Its to calculate at what time the planet is at what position.

All formulas I found are to complicated to understand.

Edit:

I want to use the formula for creating a two dimensional visualisation of the solarsystem. Objects, for example planets, should rotate around a central mass. The curve should be a ellipse and the system sould be stable.

testo
  • 111
  • 1
    Probably at issue is that the formulas you're looking at are from semi-analytic theories (e.g., VSOP, whose intended purpose is high-precision calculation of planetary positions, not conceptual understanding. It would help if you were to clarify what your intent is for this formula. What will you do with it? – Brian Tung Mar 26 '19 at 21:55
  • I added my intent to my question. – testo Mar 26 '19 at 22:01
  • Which coordinates will you use? In natural coordinates (centered at the Sun), the formula is very simple (neglecting the many-body perturbations). – user Mar 26 '19 at 22:03
  • The system would only be unstable if you were to simulate the physical laws (gravity, primarily). If you treat them as Keplerian elliipses, it's much easier. The online formulas are complicated primarily because they have to deal with planet-planet perturbations. – Brian Tung Mar 26 '19 at 22:05
  • @doev The vis-viva equation gives the speed of the planet. An ellipse has the parametric form $(a\sin t - x_1, b\cos t - y_1)$. You can use these as a starting point. – Infiaria Mar 26 '19 at 22:05
  • @Infiaria: OP may wish to have the major axis oriented at all angles, not just orthogonally. – Brian Tung Mar 26 '19 at 22:07
  • The name of the curve is ellipse, not eclipse. – user Mar 26 '19 at 22:08
  • every object should have a center were it rotates around. Earth(Sun), Moon(Earth). But I don't want to use a circle curve, cause I also want to visualize a comet. – testo Mar 26 '19 at 22:10
  • @BrianTung Simply redefine the coordinate system so that it is orthogonal then. I know you're not OP, but it doesn't seem that OP wants an excruciating detailed simulation. Planets don't travel in an ellipse anyway; only approximately. – Infiaria Mar 26 '19 at 22:14
  • @Infiaria: That only helps if all the bodies are oriented orthogonally with respect to one another. Obviously, OP does not require high precision, but it seems odd to me that they'd want elliptical orbits where all of the periapses are oriented at multiples of $90$ degrees from one another. – Brian Tung Mar 26 '19 at 22:15
  • Its not a Problem. It should be only a visual approximation of the solarsystem. – testo Mar 26 '19 at 22:18
  • @BrianTung I thought OP was talking with two bodies; in the other case, the formula would probably not be very simple. – Infiaria Mar 26 '19 at 22:18
  • I think (a sin t − x1,b cos t − y1) is nearly what I need. But the problem is, that the objects speed should be faster if it is near its central mass. – testo Mar 26 '19 at 22:22
  • @Infiaria: I thought they were, too, but then they added a comment about each body having to reference the body it was revolving around. – Brian Tung Mar 26 '19 at 22:22
  • Brian Tung, its just a geometric center. The whole simulation can be a simple fake. – testo Mar 26 '19 at 22:23
  • @doev: If each body's mass is negligible compared to its primary, then mean orbital speed is inversely proportional to the square root of the distance, and directly proportional to the square root of the primary's mass. – Brian Tung Mar 26 '19 at 22:25
  • @BrianTung, yes read about it (Kepler), but I can't build a formula with it. – testo Mar 26 '19 at 22:29
  • maybe I sould sleep one night over the question and reprase it tomorrow. – testo Mar 26 '19 at 22:31
  • 1
    @doev You said earlier that 'the problem is that the object's speed should be faster if it is near its central mass'. The vis-viva equation takes that into account (see my earlier comment). We can assume that the planet takes on a fixed path, an ellipse, so you should be able to convert speed into a velocity vector, which then gives you displacement at any time. Note that $r$ in the equation will vary, so you will need to do some calculus. Eventually, that will give you the specific equation you're looking for. I don't quite have the time to do this for you, so just ask again if you're stuck. – Infiaria Mar 26 '19 at 22:46

1 Answers1

1

If you want an approximate solution, you may use ellipses for the orbits of planets and other objects that orbit the Sun in closed paths.

We further simplify our representation and put every object in the same plane. And, as a final simplification, we will consider that the major axis of the ellipse is parallel to the $x$ axis.

Since the all objects will have the Sun in one of the focus of the ellipse, the Sun can be placed at $(0,0)$. Using this, the equation of each object can be written as: $$ x(t) = a \cos (2 \pi t / T + \theta) + \epsilon a $$ $$ y(t) = a \sqrt{1-\epsilon^2} \sin (2 \pi t / T + \theta) $$ where $T$ is the period of that object, $a$ is the semi-major axis of the ellipse, $\epsilon$ the eccentricity of the orbit, $\phi$ the angle between the major axis of the ellipse and the $x$ axis and $\theta$ the initial phase angle.

If you want to simulate the solar system, you can use the semi-major axis $a$ (in Astronomical Units, AU), the period $T$ (in years, yr) and the eccentricity (dimensionless) e.g. using the data from this site.

If you try to "simulate" some solar system objects with this equations you'll find that the planets are quite far apart from each other and that the periods are rather different.

In case you want a more accurate model, you should consider that the major axis are are not aligned for all planets (so, we would need to apply a rotation matrix to $(x,y)$. Furthermore, since the orbits are not coplanar, we should consider a 3-dimensional model and rotate each orbit plane according to its inclination to the Earth's orbit (or some other plane of reference).

Have fun!

Ertxiem - reinstate Monica
  • 1,664
  • 3
  • 11
  • 15