An unidentified spacecraft is approaching the sunny side of the Earth on a Hohma

Question

An unidentified spacecraft is approaching the sunny side of the Earth on a Hohmann Transfer from a planet somewhere in the Solar System. It has been determined that the perihelion will occur at the Earth’s orbit and the perihelion speed of the transfer orbit relative to the Sun (VSA=V2) is 41.08 km/sec. It has also been determined that the closest approach to the center of the Earth (rp) will be 9000 km. Draw a diagram of the hyperbolic flyby of the Earth by the spacecraft showing the direction of the Sun, the direction of the Earth’s motion about the Sun and shows the turning angle () and the asymptote angle (). Also draw a vector diagram of the passage?

Explanation / Answer

Any spacecraft sitting on a launch pad on Earth is already in orbit around the Sun because the Earth is orbiting around the Sun. We can put a spacecraft in orbit so it goes around the Sun the same way. From that point, the spacecraft can have its speed adjusted to change its orbit around the Sun. Those adjustments can take a craft to Mars or to Venus from Earth. This is called a Hohmann Transfer Orbit, or a least energy orbit, because this is the easiest way to send spacecraft between planets.
Basically, a spacecraft in a Hohmann Transfer Orbit is put into an irregular orbit around the Sun, with one part that is "lower" or closer to the Sun than the other. When a spacecraft is going to Mars, the spacecraft's perihelion (section closest to the Sun) will be Earth's orbit, and the aphelion (farthest distance from the Sun) will intercept the orbit of Mars at a single point. The spacecraft will change its orbit by speeding up with its thrusters. Spacecraft can go to Venus in the opposite way, with their "high" point, or aphelion at Earth, and their "low" point, or perihelion at Venus. This is done by slowing down through a retroburn, so the spacecraft can "fall" towards Venus.

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