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The time of observation after the first observation (in years), the distance between the earth and the comet (in AU) and the angle between the sun and the comet (in radians) is given.

How to find whether the comet is moving in an elliptical or hyperbolic orbit around the sun, and what the time period is (if elliptic) or the distance of closest approach to the sun (if hyperbolic). First of all, we have to find the eccentricity.

For this problem, let us make the following three simplifying assumptions:

  1. The orbit of the earth is perfectly circular with radius = 1AU and time period = 1 year

  2. The orbit of the comet is coplanar to the orbit of the earth

  3. Gravitational effects of the earth on the comet can be neglected

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    $\begingroup$ What do you mean by the time of observation after the first observation? Also, the people here really appreciate if you give what have you done and your research. $\endgroup$
    – User123
    Jun 28, 2021 at 12:39
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    $\begingroup$ If it's moving on a hyperbolic orbit, it means it's not bound, i.e. it's extrasolar. The quickest way to know whether an extrasolar object has visited us, is to check the media for interviews with Avi Loeb about aliens… $\endgroup$
    – pela
    Jun 28, 2021 at 12:48
  • $\begingroup$ The net energy on the hyperbolic orbit is positive, and on the elliptic orbit is negative. $\endgroup$
    – User123
    Jun 28, 2021 at 12:50
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    $\begingroup$ So there are 2 observations on the distance and angle done? As must be? $\endgroup$ Jun 28, 2021 at 13:25
  • $\begingroup$ @User123 that would mean taking the first observation as t=0 and in during that time, the comet covered the theta angle $\endgroup$ Jun 28, 2021 at 18:07

2 Answers 2

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This might not be so hard after all.

Below I show the math for the analytical solution for a Kelperian orbit; the catch is that it's only analytical for $t(\theta)$ and not $\theta(t)$ but that shouldn't cause a problem in this case.

I will not solve the problem for you but I'll make a recommendation how to proceed:

  • Step 1: draw a diagram on 2D paper (the best kind!) with a unit circle for Earth and a generic conic of your choice for the comet
  • Step 2: put two dots on the comet's orbit and two corresponding dots for Earth.
  • Step 3: label the Sun-Earth-Comet angles and distances for the two time points
  • Step 4: scratch your head a bit, then have fun and solve!

From this answer to What is the analytical closed-form solution of the two-body problem to verify its numerical integration results? (where you can find some python script that might help towards checking your solution numerically):

Equation 27 in Wikipedia's Kepler orbit; Properties of trajectory equation is

$$t = a \sqrt{\frac{a}{\mu}}\left(E - e \sin E \right)$$

where $a$ is the semimajor axis, $\mu$ is the standard gravitational parameter also known as the product $GM$, $e$ is the eccentricity and $E$ is the Eccentric anomaly.

The relationship between $E$ and the true anomaly $\theta = \arctan2(y, x)$ is

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

and solving for $E$:

$$E(\theta) = 2 \arctan \sqrt{ \frac{1-e}{1+e} } \tan \frac{\theta}{2}.$$

plugging back in to the first equation (but not writing it all out):

$$t(\theta) = a \sqrt{\frac{a}{\mu}}\left(E(\theta) - e \sin E(\theta) \right)$$

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  • $\begingroup$ As usual, a great answer! But what do you mesan with $t (\theta)$? Does the time depend on the angle? $\endgroup$ Jun 28, 2021 at 15:54
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    $\begingroup$ @DescheleSchilder "Below I show the math for the analytical solution for a Kelperian orbit; the catch is that it's only analytical for $t(\theta)$ and not $\theta(t)$ but that shouldn't cause a problem in this case." You need either a series approximation or a numerical technique to get the position of an object in an orbit as a function of time. $\endgroup$
    – uhoh
    Jun 28, 2021 at 16:18
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For what's it worth still after the great answer above, if you know two distances, two angles, and the time elapsed between these observed values, then in priciple you should be able to calculate the orbit of the comet wrt to the Sun. Under the given circumstances. You can transform the angle and distance from Earth to the distance and position wrt the Sun. If you know the distance and position wrt the Sun at two given times then the orbit is fixed.

Knowing only the distances or the positions at two times would be insufficient. Just as it is insufficient to know the positions of an object at two dïfferent times when it moves in the gravity field of Earth. You need the two velocities also (well, one of both at least, but at the same time, that is, two different variables). For the comet you have to make two measurements though. You have to measure two positions (angles) and two distances, as you propose.

You should be able to find out if the comet is somehow accelerated on its own (by means of a rocket controled by the members of an advanced civilization who use the comet for cover). While you may not be able to calculate the ecact orbit the knowledge of the time, positions, and distances will allow to see a divergent orbit. If they will steer in such a way to be at the same two points and the same two distances within the same time the situation would be very disturbing be cause they can change their velocity at will. It would look as if they conformed to a cone-motion but in fact they travel on a different orbit. But since they don't know the times you make the measurements, this scenario is highly improbable (not to mention the scenario of aliens in a rocket itself).

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    $\begingroup$ I think you have it almost solved! You have two legs of a triangle and the angle between them so you can get two $x, y$ points and thus two $r, \theta$ points. That's not enough to get $a, \varepsilon$ because there is an additional constant (longitude of periapsis) but with the time between the two points plus the equations in my answer I think you've got it pretty much solved. $\endgroup$
    – uhoh
    Jun 28, 2021 at 16:28
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    $\begingroup$ Time between the two points is not enough; there are always at least two Keplerian-Newtonian orbits/trajectories that go from designated point A to designated point B in fixed time t. (Admittedly, in some cases, the other way is a superluminal hyperbolic trajectory around a point-mass sun) $\endgroup$
    – notovny
    Jun 28, 2021 at 16:40
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    $\begingroup$ @notovny Rats! Well at least two solutions is better than an infinite number; $x^2 = 4$ has two solutions but we are happy with $\pm$2 and don't consider it unsolved nor unsolvable. $\endgroup$
    – uhoh
    Jun 28, 2021 at 21:48
  • $\begingroup$ @notovny What is the other orbit for the comet in this csse? The other way round? Or maybe back in time? Maybe an intermediate observation must be made. $\endgroup$ Jun 28, 2021 at 22:30

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