On October 26, 1957, a modified semiorka R-7 ballistic missile opened a page of history by launching the first satellite in history, sputnik-1. We will use this example to use the calculations related to the elliptical orbit.

#### Data

Earth’s standard gravitational parameter: mu=398 600 km^{3} s^{-2}

Average Radius of the Earth: Rt=6,378km

#### Calculations

As the altitude is not usable as such, it is converts altitudes into a radius.

Perigee radius: rp=hp+Rt=215+6,378=6,593km

Apogee radius: ra=ha+Rt=939+6 378=7 317km

We can therefore calculate the geometrical data of the orbit that are the semi-major axis a and eccentricity e.

This data is used to calculate Sputnik’s time to circle the Earth.

We can also know the velocity at each point but we will be content to calculate at the extremes of the orbit that is to say the apogee and the perigee.

With a perigee at only 215km altitude, Sputnik was braked at each pass. The velocity decreased, the semi-major axis to decrease and the orbit became circular. After 3 months the perigee velocity fell below the minimum orbital velocity at that altitude (calculated below) and could no longer maintain its orbit, it disintegrated in the atmosphere.