How calculate semi-latus rectum of parabola or hyperbola?

[Kolodez]

The semi-latus rectum is necessary for the formula

radius = semi-latus rectum / (1 + eccentricity * cos (true anomaly)).

However, if we refer to Wikipedia, we need to know the semi-major axis to calculate the semi-latus rectum. But the semi-major axis is only defined for circles and ellipses, if I am not getting it wrong.
So assume I am in a parabolic or hyperbolic orbit and I know my current location, velocity, true anomaly, specific energy, eccentricity and the periapsis radius. How can I calculate the semi-latus rectum from these values in order to be able to calculate future locations?
Thank you!

 

[n72.75]

Semi-major axis is defined for hyperbolae and parabolae.

Semi-major and semi-minor axes - Wikipedia

en.wikipedia.org

 

[dgatsoulis]

This might help: Relationships of the Geometry, Conservation of Energy and Momentum

 

[Sbb1413]

But the semi-major axis is only defined for circles and ellipses, if I am not getting it wrong.

AFAIK semi-major axis is defined for any conic sections, not just circles and ellipses, as @n72.75 has already pointed out.

 

[Kolodez]

Thanks for your replies! So if I have a hyperbola, I can also use the formula

semi-major axis = - G * major body mass / (2 * orbit specific energy)

and the semi-major axis is then negative, whereas it is positive in the case of an ellipse, right? And that yields the same formula for the semi-latus rectum, i.e.

semi-latus rectum = semi-major axis * (1 - eccentricity^2)

no matter if we have an ellipse or a hyperbola.
But in the case of a parabola, the above formulas lead to

semi-major-axis = infinity,
semi-latus rectum = infinity * 0

which makes impossible to calculate the semi-latus rectum. Did I make an error anywhere?

 

[Kolodez]

This might help: Relationships of the Geometry, Conservation of Energy and Momentum

Oh, now I saw the formula

Semi-latus rectum = 2 * periapsis radius

But to compute the right hand side, I only know the formula

periapsis radius = semi-major axis * (1 - eccentricity)

which also seems to be undefined in the case of a parabola. So I still have no idea what to do in the case of a parabola.

 

[jarmonik]

Semi-latus rectum = 2 * periapsis radius

This is wrong.

periapsis radius = Semi-latus rectum / ( 1 + eccentricity )

Parabola in orbital mechanics in the Orbiter is something that you can forget. Doesn't really exists in practice.

 

[Kolodez]

@jarmonik, this is correct if eccentricity = 1. This is what I meant, sorry.

 

[Kolodez]

I looked into the Orbiter manual once again. We have
semi-major axis = - G * major body mass / (2 * orbit specific energy),
eccentricity = sqrt (1 + 2 * orbit specific energy * h^2 / (G^2 * major body mass^2))
Hence
semi-latus rectum = semi-major axis * (1 - eccentricity^2)
= - G * major body mass / (2 * orbit specific energy) * ( - 2 * orbit specific energy * h^2 / (G^2 * major body mass^2))
= h^2 / (G * major body mass)
where h = radius x velocity. Since everything is continuous, we can use this for any eccentricity. Is this correct?

 

[jarmonik]

Since everything is continuous, we can use this for any eccentricity. Is this correct?

Yes. I don't recall any functions in orbital mechanics those would be limited to elliptical or hyperbolic orbits only, except the ones that deal with eccentric-anomaly "EcA" in which case hyperbolic versions of the functions must use hyperbolic "sin, cos" instead i.e. sinh cosh