Elliptical Orbit of Mercury

[AmalR]

Question

Given that:
- Time taken for Mercury to orbit Sun = 88 days
- Eccentricity of Mercury's orbit around the Sun = 0.206
- Average distance between Mercury and Sun = 0.387 AU
- 1 Astronomical Unit (AU) = 149, 597, 870, 700 metres

Develop a model, using parametric equations, to show the path of Mercury around the Sun.

My thinking

I've thought about Kepler's first law and how the sum of the distances from a point to both foci (one being the sun) is equal to the major axis of the ellipse. I guess this could be useful because I have the average distance between mercury and the sun, but I'm not sure how to work out the distance from the other focus to the point where mercury is at. Every other method I though of requires already knowing the length of one of the axis, so no good.

There was a tip at the end of the question saying the "average value of a function equation" could be useful- I've attached it, but I'm yet to find a use for it in the problem.


This problem's been racking my brain for a while now, hope to get some much needed help

 

[Dr.Peterson]

Not knowing the context of the question (particularly, how much you have learned about orbital motions and perhaps about differential equations, I'm not sure what they expect of you. It appears that they may want you to use the parametric equation of an ellipse with a focus at the origin, and find how the average distance from the focus relates to the major and minor axes. I don't know, though, whether they expect your parameter to be time (so that your model would accurately depict the position of the planet at any given time), or whether it only has to show the path (the curve itself) accurately.

Part of the trouble is that the "average distance" depends on the varying speed of the planet, not just on the curve, and nothing in your givens tells you that. Is this for a math class or a physics class? Are you supposed to use only the given data?

 

[AmalR]

Not knowing the context of the question (particularly, how much you have learned about orbital motions and perhaps about differential equations, I'm not sure what they expect of you. It appears that they may want you to use the parametric equation of an ellipse with a focus at the origin, and find how the average distance from the focus relates to the major and minor axes. I don't know, though, whether they expect your parameter to be time (so that your model would accurately depict the position of the planet at any given time), or whether it only has to show the path (the curve itself) accurately.

Part of the trouble is that the "average distance" depends on the varying speed of the planet, not just on the curve, and nothing in your givens tells you that. Is this for a math class or a physics class? Are you supposed to use only the given data?

Yes, it is a maths class. And we have been told orally that the parameter should be time (this would just be [MATH]\theta/88 [/MATH] though right?). We are permitted only the given data. I had an idea though, I thought I could use Kepler's First Law and substitute the average distance to Mercury from a focus as the semi-major axis. This would be because the furthest the planet can be from the sun is on the furthest vertex, and the closest it can be is the closest vertex. So when adding these distances together, the result would be the whole major axis. When dividing this by two (for the average of furthest point and closest point), the semi major axis and the average distance are both resultant.

So I guess the average value of a function equation is kind of useless...

 

[Dr.Peterson]

I suspect that the author of the problem is forgetting some details of physics, which would make this a lot harder than it may be intended to be. I think you should ask your teacher about some of your ideas, and maybe some of mine.

Planets, as I mentioned, do not move at a uniform speed, or a uniform angular velocity. (This is one consequence of Kepler's second law.) So, no, in reality time is not 1/88 of angle (in whatever unit you are using). But maybe that is what you are intended to suppose.

The average value formula will be relevant, once you figure out what motion you are to assume! Start by assuming whatever time-dependence is easiest for you to work with -- whatever parametric form you are starting with. I'll want to see what that is; perhaps you have in mind this polar form. Once you've at least started this, you'll have something to show your instructor and ask whether this is the intended assumption.

Now, I think when people talk about the average distance of a planet from the sun, they usually mean the semi-major axis, as you mentioned. I don't happen to know whether that is the actual average distance either based on the actual motion of a planet or a simplified assumption; you'll have to work that out.

Possibly some physicist here will have more to say ...

 

[topsquark]

Possibly some physicist here will have more to say ...

Yes, you are correct. I'll list out some of the details.

First you are given T, e, and the average distance from the Sun. I'm setting a to be along the semi-major axis and b the semi-minor axis, as usual. The parametric equations for Mercury's position given a time t from aphelion are \(\displaystyle ( a ~ cos(t), ~ b ~ sin(t) )\). In these coordianates the eccentricy, e, is given by \(\displaystyle e = \sqrt{ 1 - \left ( \dfrac{a} {b} \right ) ^2}\) so \(\displaystyle b = a \sqrt{1 - e^2}\).

Setting polar coordinates at the center of the system (the Sun is not, I repeat not at the center of this system! The Sun is at one of the foci.) we have that the focus, f, is \(\displaystyle f = \sqrt{a^2 - b^2}\).

To complete this we need a value for a. Here's where we use the average formula.

The distance from the Sun to Mercury is the distance as a function of time, t, is \(\displaystyle r(t) = \sqrt{(a ~ cos(t) - f)^2 + (b ~ sin(t))^2}\), so we know that the average distance from Mercury to the Sun (over one period of the motion, T) is
\(\displaystyle \overline{r(t)} = \dfrac{1}{T} \int _0^T \sqrt{(a ~ cos(t) - f)^2 + (b ~ sin(t) )^2 } ~dt\)
Solve this for a.

I don't know much about the geometry of ellipses. Maybe there's a way to simplify that monster that I haven't seen. But it can be done in principle. (Though you might have to use an approximation method.)

-Dan

Addendum: Hmmmm... t is representing the angle, not the time, so the integral should be
\(\displaystyle \overline{r(t)} = \dfrac{1}{2 \pi} \int _0^{2 \pi} \sqrt{(a ~ cos(t) - f)^2 + (b ~ sin(t) )^2 } ~dt\)

This will still give you a. From there you need to work out the period T in terms of a. This isn't too bad. I leave it to the OP to finish off. (Hint: Think of Kepler's laws.)