Hohmann transfer orbit (Earth-Jupiter system)

Question

Consider the Hohmann transfer orbit of a probe launched from Earth to Jupiter. Assume the orbits of Earth and Jupiter are circular.

I want to calculate the following:

1. The semi-major axis and eccentricity of Hohmann transfer orbit from Earth to Jupiter.
2. The speed of the probe at Earth.
3. The launch speed of the probe from Earth to reach Jupiter.
4. The duration of the flight to Jupiter.
5. The elongation of Jupiter at launch to ensure that the probe arrives at Jupiter.

Here my attempt:

1. $a = \frac{1}{2}(r_E + r_J) = \frac{1}{2}(1+ 5.203) = 3.102 \text{ AU}, \ \ e = 1 − \frac{r_E}{a} = 0.677$

2. I am thinking of using the vis-viva equation but I don't know what I should use for $r$? Should I use $r_E$ = 1 AU for $r$ where $r_E$ is the average distance from the Earth to the Sun? Am I right? Please correct me if I am wrong.

3. I am thinking of using the escape velocity formula where $M$ is the mass of the Earth and $R$ is the radius of the Earth. Please correct me if I am wrong.

4. $t = \frac{1}{2}P = \frac{1}{2} \sqrt{a^3} = 2.73 \text{ years}$.

5. I really don't know how to do this question. Please help.

Answer 1

I want to calculate the following:

1. The semi-major axis and eccentricity of Hohmann transfer orbit from Earth to Jupiter.
2. The speed of the probe at Earth.
3. The launch speed of the probe from Earth to reach Jupiter.
4. The duration of the flight to Jupiter.
5. The elongation of Jupiter at launch to ensure that the probe arrives at Jupiter.

1. $a_E$, $a_J$ are 1 and 5.2 AU, respectively. The Hohmann transfer ellipse touches both, so the major axis is 6.2 AU and the semi-major axis is half of that, or 3.1 AU as you wrote.

2. Definitely use the vis-viva equation whenever you can! The standard gravitational parameter $GM$ of the Sun is 1.327E+20 m^3/s^2. You can use that if you convert your distances to meters to get the velocity in m/s,

   $$v^2 = GM\left( \frac{2}{r} - \frac{1}{a} \right)$$

   or as long as you are orbiting the Sun you can stick with AU and years and use:

   $$v^2 = \frac{2}{r} - \frac{1}{a}$$

   which will give the velocity in AU per year (once you take the square root).

3. I'm not exactly sure what "The launch speed of the probe from Earth" means and how it is different from the speed of the probe at 1 AU. If it means the speed you'd fire it from the surface of the Earth from a gun if there was no air, then just add the Earth's escape velocity to the velocity from #2.

4. this answer should help you calculate the period of an orbit for a given semi-major axis . For the Hohmann transfer ellipse, use its semi-major axis to calculate its period, and then use half of the period for the duration of the flight from Earth to Jupiter.

5. I think you can figure this one out for yourself. Draw a picture of the 1 AU and 5.2 AU circles and the Hohmann ellipse that touches both. Put a dot where you want Jupiter to be when you get there, then move Jupiter backwards by a certain angle, which would be 360 degrees times the ratio of the flight time to Jupiter's period. If the flight time is 5 years (it's not) and Jupiter's period is 10 years (it's not) then you'd back Jupiter up by 180 degrees for its starting position when you launch from Earth. Then read how elongation is defined.