Calculate foci of planetary orbit given known parameters

[CarlosMarti12]

Hello everyone. I am trying to calculate the focal points of the ellipse traced by a planet in orbit around a star, given the following known parameters:

• $M_{sun}$ Mass of the sun
• $r_{planet}$ Position vector of the planet from the sun
• $m_{planet}$ Mass of the planet
• $v_{planet}$ Velocity of the planet
• $G$ Gravitational constant
• $F_{grav} = G\frac{Mm}{r^{2}}$

I would like to find (if possible) the orbital information of the planet (including eccentricity, focal points, and axes, if possible) based on these pieces of data. Is it possible to calculate the focal points of its orbit based on these parameters? If more parameters are necessary, I might be able to add them. Any help would be greatly appreciated! (The focal points of an ellipse are shown below.)

 

[lippyka]

Focal Points: (c, 0), (-c, 0)Where c is the distance between the two focal points.Yes, it is indeed possible to calculate the focal points of the ellipse given the above parameters. The eccentricity of an orbit can be calculated from the following equation:e = \sqrt{1 + \frac{2EL^{2}}{m_{planet}G M_{sun}}Where E is the total mechanical energy of the system and L is the angular momentum of the planet.Once the eccentricity is known, the focal points can be calculated using the following formula:c = a\sqrt{1 - e^{2}}Where a is the semi-major axis of the orbit.The semi-major axis is given by the equation:a = \frac{r_{planet}v^{2}}{G M_{sun}}Combining all these equations, we get:c = \frac{r_{planet}v^{2}}{G M_{sun}}\sqrt{1 - \left(1 + \frac{2EL^{2}}{m_{planet}G M_{sun}}\right)^{2}}Therefore, given the parameters specified in the question, it is possible to calculate the focal points of the ellipse traced by the planet in its orbit around the star.