## Info

Earth when Probe Reaches Venus

Earth at Launch

Earth at Launch Earth when Probe Reaches Venus

Minimum energy orbit of a space probe to

Earth's

Fig 22.15.

### Venus.

Minimum energy orbit of a space probe to of the distance of the Earth from the Sun and the distance of Venus from the Sun. Therefore, the semi-major axis is a = (1/2 )(Re + Rv )

The eccentricity is defined by the separation between the foci, divided by 2a. The Sun is at one focus, RV from perihelion. Therefore, the symmetry of the orbit requires that other focus to be RV from the aphelion. The distance between the foci is therefore d = 2a - 2RV = Re + Rv — 2Rv = re — RV The eccentricity is therefore should be noted that 0 = 90° doesn't give a circular orbit unless the speed is the right speed for a circular orbit at that radius. However, 0 = 90° means that the point where the fuel is spent will be the aphelion or perihelion of the orbit.

The trick in launching a planetary probe is to place it in an orbit that intersects the planet's orbit when the planet is at the point of intersection. In choosing the direction of launch, we can take advantage of the Earth's motion. There are many different orbits that can be chosen. However, the minimum energy orbit is the one that has the Earth at aphelion and the planet at perihelion (if the planet is closer to the Sun) or one that has the Earth at perihelion and the planet at aphelion (if the planet is farther from the Sun). The minimum energy orbit of probe to Venus is shown in Fig. 22.15.

Example 22.3 Minimum energy orbit to Venus Find the semi-major axis and eccentricity of a minimum energy orbit from the Earth to Venus. Find the necessary launch speed and the time for the trip.

solution

In this case the Earth is at aphelion and Venus is at perihelion. The major axis of the orbit is the sum

At launch, r = RE so the launch speed is given by v2 = GMn I — -