How to find the velocity vector of a planet in orbit argument of periapsis or longitude of the ascending node?

Question

Edit : This question has been answered. If anybody else creating a fictional planet has a similar question to mine, with a not-so-good understanding of orbital mechanics such as me, the answer should review where I went wrong and what to do in a similar situation. Thanks to those who put the effort into helping me understand.

For a personal project, I am creating a fictional planet using as much actual science as possible. As the title suggests, I only have three of the orbital elements and wish to calculate the velocity vector (which I am having trouble with), so that I can calculate the specific relative angular momentum vector. As far as I understand, this vector is required for calculating the argument of periapsis and longitude of the ascending node. Following is the list of relevant measures I HAVE calculated for my planet :

Keplerian Elements :

• e = 0.6590606759
• θ (true anomaly) = 1.446411718 rads
• a = 1.957694954 AU
• i = 0.2316051917 rads

Other descriptors derived from the above :

• b = 1.472362352 AU
• e = (0.6590606759,0,0)
• r (distance between bodies) = 1.02364918873 AU
• r = (0.1697602404,0.981,0.238071674877)
• v (current position) = 38975.36618 m/s
• E = 0.7613114023 rads
• M = 0.3066442884 rads

The coordinates of the planet and star in AU are (respectively) (1.46, 0.981, 0.232571546481), (1.29023975963, 0, 0).

I will consider epoch to be the periapsis, so that t0=0 when r=0.66745519437 AU and v=51153.89281 m/s. The time since the periapsis passage is t1=3872747.757 seconds.

If any formulas are given as answers, it would be helpful if an example of use is given with said formula, as this is how best I learn. You, by no means, need to do this, as I will likely figure out the formulas in time.

Sorry if there are any, or many, egregious mistakes, in either formatting, specificity, or calculations. This is both my first time asking here, and using orbital mechanics/linear algebra. Any additional information/clarification can be added if needed. Thanks for your time.

Answer 1

Because this is a fictional planet, the only thing equations can tell you is whether the choices you have made are compatible with each other. In this case, you need to make a few more choices before we can say whether the whole scenario works.

The first thing to keep in mind, even in the idealized two-body problem with no other planets, moons, or complications, is that some of these quantities are constant while others change with time. The orbital elements which are approximately constant for short times, so that you can pick just one number for each to describe the whole orbit, are:

• $a$, semimajor axis (distance)
• $e$, eccentricity (unitless)
• $i$, inclination (angle)
• $\Omega$, right ascension of the ascending node (angle)
• $\omega$, argument of periapsis (angle)

All five of these can be chosen independently of the others. That means you have to choose $\Omega$ and $\omega$, to be whatever you want, before you can have a complete description of the orbit.

Choosing $\Omega$=0, and also $i$=0, is natural for the planet around its star, because in real-world astronomy, that's how the ecliptic is defined. The astronomers on your world would probably make the same choice: their planet's relationship with its sun is the standard used to measure everything else against.

Choosing $\omega$=0 means the time of closest approach is the same as the start of your time scale. Any other value just means the zero point in time, as measured by $M$, happens somewhere other than the zero point in angle, measured by $\theta$, which is zero at periapsis. For the purposes of your fiction, the interesting choice is where is the planet in its orbit ($\theta$) when the calendar year begins.

The eccentricity vector is composed of these, so it also stays constant in time. The first consistency condition we find is that while $a$, $b$, and $e$ are all constant, you can only choose two of them, because of the equation $$e = \sqrt{1 - \frac{b^{2^\phantom{A}}}{a^2}}$$

Everything else is changing continuously. As time in your story passes, the planet moves around its orbit. The three anomalies and the position and velocity vectors all describe where on the orbit the planet is at one particular instant. What the constant values allow you to do is figure out where on its orbit ($\theta$) the planet will be at some other time ($M$) in its year.