In apogee or perigee?

The spacecraft starts from the an artificial Earth satellite, moving around the planet in an elliptical orbit. At what point is it more profitable to launch - when the satellite will be in apogee or perigee? It would seem that the answer is perfectly clear: of course, at apogee: after all, the farther from Earth, the weaker the gravity of gravity, the lower the release rate, and, consequently, the less necessary fuel consumption. However, it should not be forgotten that, according to Kepler's second law, the satellite is moving in its orbit with a variable speed. And at apogee it is the lowest, and at the perigee is the highest. What is more profitable? A lower release rate at apogee, but also a smaller reserve of initial speed or a greater margin of initial speed at perigee, but also a higher release speed, which the ship should gain? No qualitative considerations will give an answer to this question - exact calculations are needed. It is necessary to calculate, for apogee and perigee, the difference between the speed of movement of an artificial satellite and the speed of liberation at a given point near-earth space and compare these differences with each other. Obviously, preference will be given to the variant of launching an artificial satellite, for which this difference will be smaller.

The spacecraft starts from the an artificial Earth satellite, moving around the planet in an elliptical orbit

Let's consider a concrete example. Let the launch of the spacecraft be carried out from the side of an artificial satellite of the Earth, which moves in orbit with a height of apogee 330 km and a perigee altitude of 180 km. The values of the release rate for different altitudes have long been calculated and summarized in special tables. Looking at this table, we find that for the perigee altitude of the orbit of this satellite of the Earth it is 11040 m/s, and for the altitude of apogee 10918 m/s. It is not difficult to calculate the speed of the satellite's motion in perigee and apogee. It is 7850 and 7680 m/s, respectively. Now calculate the required differences. For the perigee 11040 - 7850 = 3190 m/s, for the apogee 10918 - 7680 = 3238 m/s. Thus, the more advantageous point for a hundred is not apogee, as it might seem at first glance, but perigee. Curiously, with the increase in the ellipticity of the orbit, the advantages of a perigee start increase to an even greater degree and the paradoxicality of the situation becomes particularly clear. For example, with a very elongated orbit with perigee at a distance of 40 thousand km from the Earth and apogee located beyond the lunar orbit at a distance of 480 thousand km from our planet, to reach the second cosmic velocity and escape from the "clutches" of the earth's gravitation in four (!) times more easily from the perigee area than from the apogee area. It's strange, is not it? This fact once again demonstrates the deceptiveness of many visual representations. However, it should be stressed once again that the paradox in question is only valid when comparing the benefits of launches from the same satellite moving in a given orbit. It is interesting that when the artificial satellite of the Earth decreases, a reverse paradox takes place. It would seem that it would be more advantageous to include a braking motor system and begin braking at the moment when the satellite passes the perigee, i.e., is closest to the earth's surface. But calculations show that even in this case the main role is played not by the distance from the Earth, but by the speed of the satellite's motion along the orbit. In the apogee, it is lower, and therefore, from the point of view of fuel consumption, descent is most expedient from the apogee portion of the orbit. True, in this case we are talking about a somewhat idealized problem, since the speed of the satellite's entry into the dense layers of the earth's atmosphere is not taken into account.