Calculate elliptical orbit using attitude and velocity

[donggas90]

Hi. I'm just a hobbier of astronomy and have a question about elliptical orbit.
I wonder that can calculate elliptical orbit using just atitude(location) and velocity(vector).

Please look at my picture.
The blue dot is central body and green dot is my interesting body.
Let me assume mass of central body is enough huge to ignore green's gravity.
and assume that there are only those two objects in universe(prevent multi-body problem).
Then we know that the blue dot's mass, gravity and green dot's mass, location and velocity.

I want to get the purple line's length, semi-major axis,
and the orange line's length, semi-minor axis what can calculate from semi-major axis using eccentricity. Finally location of perihelion or aphelion.
Already have seen many questions and answers about calculating orbital speed(scalar) from atitude using vis-viva but velocity(vector).
I think, it's too hard to predict where will be perihelion or aphelion.

How can I solve this problem?

 

[Nugatory]

It is indeed possible to calculate the orbit from the information you have - This was the first great triumph of classical mechanics more than three centuries ago. You'll find how it's done if you google for "Planetary motion equations" (some math background required - elementary differential equations and comfort with polar coordinates).

 

[donggas90]

It is indeed possible to calculate the orbit from the information you have - This was the first great triumph of classical mechanics more than three centuries ago. You'll find how it's done if you google for "Planetary motion equations" (some math background required - elementary differential equations and comfort with polar coordinates).

Thanks for answer.
In additionally, I had been successfully calculated location of planet in a moment using given major axis and eccentricity from this article what you said.
However, cannot apply that topic to this problem. because that topic do not use any velocity but position and aphelion that had been decided.
Can you give me some more hints?

 

[Nugatory]

Thanks for answer.
In additionally, I had been successfully calculated location of planet in a time using given major axis and eccentricity from this article what you said.
However, cannot apply that topic to this problem. because that topic do not use any velocity but position and aphelion that had been decided.
Can you give me some more hints?

Although this problem was solved some centuries ago, it's by no means trivial. The best advice that I can give you is to get hold of a college-level mechanics textbook (I learned from Kleppner and Kolenkow, but others here will doubtless have their own favorites) and work through the section on central-force motion.

 

[phyzguy]

The Wikipedia page on Kepler orbits basically tells how to do this. In particular the part labeled "http://en.wikipedia.org/wiki/Kepler...bit_that_corresponds_to_a_given_initial_state

It fairly easy to calculate the perihelion and aphelion distances if you remember that the orbit has only two constants of the motion, the total energy and the total angular momentum. So if you calculate these two quantities (L , E) for your given initial conditions, they are constant all along the orbit. At perihelion and aphelion, the velocity vector is perpendicular to the radius vector, so at these points the angular momentum L is just m*r*v, and the energy E = m*v^2/2 - G*M*m/r, so given E and L you have two equations in two unknowns to calculate r and v. This gives you the perihelion distance Rp and the aphelion distance Ra, from which you can calculate the semimajor axis a and the eccentricity e.

 

[donggas90]

Although this problem was solved some centuries ago, it's by no means trivial. The best advice that I can give you is to get hold of a college-level mechanics textbook (I learned from Kleppner and Kolenkow, but others here will doubtless have their own favorites) and work through the section on central-force motion.

OK, now I got that book. I'll reference that part. Thanks again.

 

[donggas90]

The Wikipedia page on Kepler orbits basically tells how to do this. In particular the part labeled "http://en.wikipedia.org/wiki/Kepler...bit_that_corresponds_to_a_given_initial_state

It fairly easy to calculate the perihelion and aphelion distances if you remember that the orbit has only two constants of the motion, the total energy and the total angular momentum. So if you calculate these two quantities (L , E) for your given initial conditions, they are constant all along the orbit. At perihelion and aphelion, the velocity vector is perpendicular to the radius vector, so at these points the angular momentum L is just m*r*v, and the energy E = m*v^2/2 - G*M*m/r, so given E and L you have two equations in two unknowns to calculate r and v. This gives you the perihelion distance Rp and the aphelion distance Ra, from which you can calculate the semimajor axis a and the eccentricity e.

Oh, I just missed about that. Thank you.

 

[Nugatory]

The Wikipedia page on Kepler orbits basically tells how to do this. In particular the part labeled "http://en.wikipedia.org/wiki/Kepler...bit_that_corresponds_to_a_given_initial_state

It fairly easy to calculate the perihelion and aphelion distances if you remember that the orbit has only two constants of the motion, the total energy and the total angular momentum. So if you calculate these two quantities (L , E) for your given initial conditions, they are constant all along the orbit. At perihelion and aphelion, the velocity vector is perpendicular to the radius vector, so at these points the angular momentum L is just m*r*v, and the energy E = m*v^2/2 - G*M*m/r, so given E and L you have two equations in two unknowns to calculate r and v. This gives you the perihelion distance Rp and the aphelion distance Ra, from which you can calculate the semimajor axis a and the eccentricity e.

I already "liked" this reply, but should add that it's a much better answer than mine. The question doesn't require all the machinery of a general orbital solution.