How to determine the delta V required in order for the satellite at pt. A to hit the earth at pt. B?

Question

I'm confused about how would I go about obtaining delta-v in this case. I'm aware of orbital transfers such as bi-elliptical but that is not the answer to this. Kindly help

Answer 1

The answer is much depending on how realistic you try to model "your world" and how small your error margin (are you trying to hit a "POINT" or an "AREA"?) is. Do you need the solution for minimum delta V or "a possible" delta V?

Most simple: reduce it to a 2D scenario with no atmospheric drag - simple hohmann transfer -> deltaV for R_orbit to R_earth

Most (possible) realistic: you first would need to find out if your orbit is overflying the target once a while. If not (to small inclination) you first would need to change your inclination (costly in terms of delta V). Then wait for a chance where your S/C is overflying the Target and calculate a retroburn shortly (about half an orbit) before the overflying.

In fact you would need a numerical propagator and kind of recursive approach. Your S/C is mostly influenced by the atmosphere during this time unfortunately the atmospheric models are not perfect so you will end up with an uncertainty. This is okey, as long, as you have a target area big enough.

Summed up: no simple formula for a short answer.

Another possibility would be a kind of active controlled path. So you would correct your decent path with plenty of maneuvers (e.g. like ballistic missiles) but this cost you fuel.

Answer 2

I interpret this as "minimum delta-v as a function of an arbitrary elliptic orbit and an arbitrary point on the surface".

Furthermore, as a pure orbital mechanics problem, I assume the Earth to be perfectly spherical and ignore the atmosphere. However, I will still allow the Earth to rotate.

The general case can always be solved by increasing apogee up to infinity, and from there adjust inclination to lower the perigee to hit any point.

Since the cost of those high apogee manoeuvrers are approaching zero, the total cost is:

$$\Delta v = \sqrt{\frac{2\mu}{r_P}} - \sqrt{\mu\left(\frac{2}{r_P} - \frac{2}{r_P + r_A}\right)}$$

It is sometimes not possible to do better than this.

In the planar case, it's sometimes less costly to lower the periapsis directy, after waiting some appropriate time for B to align right (and if the orbital period is some simple fraction of a rotational period, it can be fudged by a burn approaching zero cost).

$$\Delta v = \sqrt{\mu\left(\frac{2}{r_A} - \frac{2}{r_P + r_A}\right)} - \sqrt{\mu\left(\frac{2}{r_A} - \frac{2}{r_{earth} + r_A}\right)}$$

If they are not coplanar, optimal inclination change is non-trivial even for the 2-body problem