Calculate eccentricity from apside altitudes? I need to calculate the eccentricity of a satellite orbit with only the apside altitudes (height above the surface of the Earth), 400 and 1500 km. However, I don't know the semi-major axis ($a$) or semi-minor axis ($b$).

$$\text{eccentricity} = (1-(b/a)^2)^{1/2}$$
The major axis is just the sum of the distance at perigee plus the distance at apogee. This is 400 km + 1500 km + the diameter (not radius) of the Earth, and the semi-major axis is half of that.

However, the semi-minor axis is harder to determine. Therefore, why not use an alternative method to calculate the eccentricity?

$$e = \frac{r_a-r_p}{r_a+r_p}$$

Here, the $r$'s are the distance to the center of the Earth, not the apside altitudes. Can you take it from here?