1
$\begingroup$

Question An ellipse E is arbitrarily oriented in 3-D space with the origin at one of its foci. Use the standard elements provided in the figure. Assume the reference direction $\gamma$ is the positive Y-axis direction. Find the projected position vector of the ellipse onto the plane of reference as a function of the angle $\nu$.

After some calculation, I obtained the position vector of the celestial body to be $$r=\frac{a(1-e^2)}{1+e\cos(\nu)}$$ Where $a$ is the semi-major axis and $e$ is the eccentricity. Now, I begin with the $(r,\nu)$ coordinates in the plane of the ellipse and project it onto the reference plane using the $\cos(i)$ factor. How, then, can I find the orthogonal components $(x,y)$ of this projected position vector on the reference plane? A parameterized solution would be appreciated as well.

$\endgroup$

1 Answer 1

1
$\begingroup$

You just need to find a unit vector $\hat n$ giving the direction of the planet, and then multiply it by $r(\nu)$ given by your formula.

Suppose then your planet is at $(0,1,0)$: to carry it to its actual position $\hat n$ you need to perform first a rotation about $z$-axis by an angle $\omega+\nu$, then a rotation about $y$-axis by an angle $i$ and finally a rotation about $z$-axis by an angle $\Omega$: $$ \hat n= \pmatrix{\cos\Omega &-\sin\Omega & 0\\ \sin\Omega &\cos\Omega & 0\\ 0 & 0 & 1}\cdot \pmatrix{\cos i & 0 &\sin i\\ 0 & 1 & 0 \\ -\sin i & 0 &\cos i}\cdot \pmatrix{\cos(\omega+\nu) &-\sin(\omega+\nu) & 0\\ \sin(\omega+\nu) &\cos(\omega+\nu) & 0\\ 0 & 0 & 1}\cdot \pmatrix{0\\ 1\\0}, $$ giving the result: $$ \hat n= \pmatrix{-\cos i \cos\Omega \sin(\omega+\nu)-\sin\Omega \cos(\omega+\nu)\\ \cos\Omega \cos(\omega+\nu)-\cos i \sin \Omega\sin(\omega+\nu)\\ \sin i \sin(\omega+\nu)}. $$ Multiply that by $$ r(\nu)=\frac{a(1-e^2)}{1+e\cos(\nu)} $$ and you'll have the coordinates of the planet.

$\endgroup$
16
  • $\begingroup$ Okay, so I tried to follow your logic; I understand the first rotation, but how is the second rotation about the Y axis? And isn't the third rotation about the line passing through the origin and perpendicular to the plane of the ellipse? $\endgroup$
    – Arnav Das
    Jun 24, 2020 at 15:40
  • $\begingroup$ The second rotation is about the new position of $y$-axis after the first rotation, and so on. $\endgroup$ Jun 24, 2020 at 15:44
  • $\begingroup$ @ArnavDas Try with some simple values of the angles, to see if you get the right coordinates. $\endgroup$ Jun 24, 2020 at 15:44
  • $\begingroup$ Won't you have to change the matrix to rotation around the "new" Y axis? Something seems to be wrong here. Do you mind having a look at this? colab.research.google.com/drive/… $\endgroup$
    – Arnav Das
    Jun 26, 2020 at 4:29
  • $\begingroup$ When you rotate a vector by i, you need to do it not around the old Y axis but the new Y axis. The new Y axis is the unit vector rotated by Omega around Z. Do you know how to form the rotation matrix for such an axis? $\endgroup$
    – Arnav Das
    Jun 26, 2020 at 4:39

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .