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I have to solve a rather extensive task.

**Most important are questions a) and c).**I appreciate any help :)!

## Homework Statement

The task: Determine characteristics of a Hohmann transfre towards, when circular planetary orbits around the Sun with [itex]R_E = 149.6 \cdot 10^6 km[/itex] for Earth and [itex]R_M = 227.9 \cdot 10^6 km[/itex] for Mars are used.

The task:

**a)**How does the true anomaly in

*[deg]*of the Earth's position between start and arrival of the satellite change?

**b)**What is the angle

*[deg]*between Earth / Sun/ Mars when the satellite arrives at Mars?

**c)**How many days before opposition (Earth is on a line between Sun and Mars) the launch from Earth must occur to arrive at the apocentre of the transfer ellipse, when Mars is to be encountered.

## Homework Equations

In different sub-tasks I was already able to identify following.

The orbit transfer time [itex]t_H = 258.8 [days][/itex]

The orbital velocity of Earth [itex]\nu_E = \sqrt{\frac{\mu_{Sun}}{R_E}} = 107224.1113 \frac{km}{h}[/itex]

The orbital velocity of Mars [itex]\nu_M = \sqrt{\frac{\mu_{Sun}}{R_M}} = 86870 \frac{km}{h}[/itex]

The angular velocity of Earth [itex]\omega_E = \frac{360^\circ}{365[days]} = 0.9863\frac{deg}{days}[/itex]

The angular velocity of Mars [itex]\omega_M = \frac{360^\circ}{686.795[days]} = 0.5242\frac{deg}{days}[/itex] (received from the orbital period T).

## The Attempt at a Solution

**a)**For the true anomaly I only found a formula covering the "One-tangent burn" orbit changing technique: [itex]\nu = \arccos{(\frac{a_{tx}(1-e^2)/r_B-1}{e})}[/itex]. However I do not know, if I can apply this formula to a Hohmann transfer.

**b)**The satellites travels [itex]180^\circ[/itex] in [itex]258.8 [days][/itex] in the same time. The earth travels [itex]255.2544^\circ[/itex] and Mars travels [itex]135.6629^\circ[/itex]. Since Mars only performs [itex]135.6629^\circ[/itex] we must have given him a headstart of [itex]44.3371^\circ[/itex]. So the angle at start is [itex]0^\circ[/itex] between Sun and Earth and [itex]44.3371^\circ[/itex] between Sun and Mars. At end the angle between Sun and Earth is [itex]255.2544^\circ[/itex] and [itex]180^\circ[/itex] between Sun and Mars.

**Is this correct?**

**c)**From b) I know that Mars needs a headstart of [itex]44.3371^\circ[/itex]. Hence I was able to calculate how many days must have been passed until Mars is in the opposition of earth.

[tex]\omega_E \cdot x = 44.3371 + \omega_M \cdot x \Leftrightarrow x = 94.48 [days][/tex]

After 94.48 days after start Mars opposes Earth.

**In addition I have no idea how to solve this task :).**

__Thank you very much for your time!__