Question

A satellite has these orbital elements: -- semi-major axis = 10,900 km -- eccentricity = 0.1 -- inclination = 60 degrees -- longitude of the ascending note = 175 degrees -- argument of perigee = 90 degrees -- true anomaly = 90 degrees

Where is the satellite located in its orbit?

Answer 1

A space vehicle's orbit may be determined from the position and the velocity of the vehicle at the beginning of its free flight. A vehicle's position and velocity can be described by the variables r, v, and , where r is the vehicle's distance from the center of the Earth, v is its velocity, and is the angle between the position and the velocity vectors, called the zenith angle (see Figure 4.7). If we let r₁, v₁, and ₁ be the initial (launch) values of r, v, and , then we may consider these as given quantities. If we let point P₂ represent the perigee, then equation (4.13) becomes

Substituting equation (4.23) into (4.15), we can obtain an equation for the perigee radius R_p.

Multiplying through by -R_p²/(r₁²v₁²) and rearranging, we get

Note that this is a simple quadratic equation in the ratio (R_p/r₁) and that 2GM /(r₁