Undergrad A little doubt regarding specific angular momentum

[cptolemy]

Good afternoon

I just have this little doubt: imagine the osculating orbit of Mars changing slowly in its elements along the centuries. The semi major axis changes, the period, etc.

Is the specific angular momentum allways equal in all the osculating orbits Mars has in those centuries? Or does it changes with the elements change? That is, if in a century the orbital semi major axis is 1.2 greater, will the specific angular momentum be the same as the previous?

Is it constant on one 2 bodies orbit even if the semi major axis increases?

Kind regards

JKepler

 

[gneill]

Is the specific angular momentum allways equal in all the osculating orbits Mars has in those centuries?

I would think that it could change, depending upon the details of the change in orbit. Interacting bodies can exchange both linear and angular momentum.

Consider the expression relating specific angular momentum and the size of the latus-rectum:
$$p = \frac{h^2}{\mu}$$
If the orbit changes affect the latus-rectum, then h must change accordingly.

 

[cptolemy]

Good evening

Yes, I agree. Thanks for the feedback. However, in 2 different orbits of the same planet, will the same amount of area done in equal times intervals? The total areas and periods are different. But is the second law applyed, and these two values proportional?

Clear skies

JKepler

 

[gneill]

If you add or subtract energy from an orbit (rotational or angular) you can expect changes. Kepler's laws for 2-body don't take into consideration the influence external influences.

So I would say no, the laws don't apply between different osculating orbits.

Take an extreme hypothetical example. An external body collides with a planet and reduces its orbital velocity to nil (ignore the unphysical technicalities). After this the area swept out per unit time would be nil.

 

[cptolemy]

Thanks for the reply gneill

So hipothetically, if a body with velocity V in a orbit of semi major axis A with an eccentricity of E, when it's at its perigee:

dist = A(1-E) and velocity
V= √μ/A*(1+E)/(1-E)

is reduced to a velocity V' by some momentanious force outside, what will be the new parameters of the orbit (the new A' and E')? How do I translate the velocity variation ΔV (V' - V) in those elements?

Can you help me? Thanks.

JKepler

 

[gneill]

Given the velocity and distance I'd start by finding the specific mechanical energy ξ for the new orbit, hence the new semimajor axis. Then, if you've got both the radius and velocity as vectors after the change, find the new specific angular momentum $\vec{h} = \vec{r} \times \vec{v}$. With $\vec{v}\;,\;\vec{r}\;,\text{ and } \vec{h}$ in hand you can find the eccentricity vector.

Note that this is not dissimilar to the Hohmann Transfer situation where a spacecraft intentionally changes its velocity to put it on an intercept orbit for another orbit. The only difference is we don't have the second velocity change to circularize the orbit at the new distance.