Parameterization of an ellipse

Question

If an object (like a planet) orbits around a more massive object (like the sun) the orbit will be an ellipse with the massive object at one of the two foci of the ellipse. The parameterization $$x(t) = 2 \cos(t), \text{ and } \ y(t) = \sin(t)$$ is a parameterization of the ellipse $$\frac{x^2}4 + y^2 = 1,$$ which has foci at the points $(−\sqrt 3 , 0)$ and $(\sqrt 3 , 0)$.

Could this parameterization be a parameterization of an object in orbit? Explain why or why not.

I believe the answer is yes this parameterization could be one of an object in orbit, however the only reason i think that is because sin and cos form an ellipse that looks like it could rotate around a planet.

I am not concrete in my reasoning. Would appreciate any sort of help that could help me wrap my head around this problem to understand it a bit better. Thanks

Answer 1

What you have given is a path, an orbit without reference to time or acceleration.

Take a simpler example like Newton did some 3 hundred years ago.If you have a vertical line you can come down at constant speed or constant acceleration or in any of myriad ways you like to choose, varying these as functions of time and height.

If $t$ represents a polar coordinate angle of ellipse, inverse square law force acting at $ (-\sqrt 3 ,0)$ makes it possible to move as a planet along an ellipse in a unique way, by conserving product of radius and velocity product.

Your parameterization is correct in space but it entirely lacks the dynamic picture. It defines what you want but does not hint how it is brought about,or by what rules with respect to time such a motion can be brought about.

But if you say

$$ p = \sqrt{x^2+y^2} - \epsilon \cdot x $$

and define constants $ p, \epsilon $ in a certain force equilibrium ( latus rectum, eccentricity) set up it could be fully valid as for a planet speeding when nearby and slowing down when farther away.

Answer 2

The equations below describes well the shape of an orbit but not the motion of the orbiting object; $t$ does not have anything to do with time. $$x(t) = 2 \cos(t), \text{ and } \ y(t) = \sin(t),\tag 1$$ $$\frac{x^2}4 + y^2 = 1.\tag 2$$

If you consider $t$ to be the time then $(1)$ would describe a motion uniform in time along the ellipse. But we know that the motion of the orbiting planets is not uniform. So this motion does not have to do with Newtonian mechanics.

If I am not mistaken the OP would like to see a parametrization that decribes the Newtonian motion.

In wikipedia, you can find a motion picture showing the movement of an orbiting object around the sun the Newtonian way. (A Feynman lecture) Based on the idea given in the wiki article we can make up a right parametric description of the planets.

In the following figure a hypothetical object (a red Angel) orbits around the origin uniformly in time along a circle:

Here $t$ reflects to time. The shape of the orbit of the planet is given by

$$\frac{(x+\sqrt[3]3)^2}4+y^2=1.$$

(One of the foci in $(2)$ has been shifted to the origin.) The red line connects the uniformly orbiting red Angel with the origin. The red line crosses the ellipse at the location of the planet. As the Angel orbits uniformly, the planet moves along the ellipse but not uniformly.

One can get the time-parametric equation of the planet by determining the $x(t),y(t)$ coordinates of the red point. This is an easy task given that the equation of the red line is

$$y=\tan(t)x.$$