<blockquote>
  <p>I want to calculate the following:</p>
  
  <ol>
  <li>The semi-major axis and eccentricity of Hohmann transfer orbit from Earth to Jupiter.</li>
  <li>The speed of the probe at Earth.</li>
  <li>The launch speed of the probe from Earth to reach Jupiter.</li>
  <li>The duration of the flight to Jupiter.</li>
  <li>The elongation of Jupiter at launch to ensure that the probe arrives at Jupiter.</li>
  </ol>
</blockquote>

<ol>
<li><p><span class="math-container">$a_E$</span>, <span class="math-container">$a_J$</span> are 1 and 5.2 AU, respectively. The <a href="https://en.wikipedia.org/wiki/Hohmann_transfer_orbit" rel="nofollow noreferrer">Hohmann transfer ellipse</a> 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.</p></li>
<li><p>Definitely use the <a href="https://en.wikipedia.org/wiki/Vis-viva_equation" rel="nofollow noreferrer">vis-viva equation</a> whenever you can! The <a href="https://en.wikipedia.org/wiki/Standard_gravitational_parameter" rel="nofollow noreferrer">standard gravitational parameter</a> <span class="math-container">$GM$</span> 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, </p>

<p><span class="math-container">$$v^2 = GM\left( \frac{2}{r} - \frac{1}{a} \right)$$</span></p>

<p>or <strong>as long as you are orbiting the Sun</strong> you can stick with AU and years and use:</p>

<p><span class="math-container">$$v^2 = \frac{2}{r} - \frac{1}{a}$$</span></p>

<p>which will give the velocity in AU per year (once you take the square root).</p></li>
<li><p>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.</p></li>
<li><p><a href="https://astronomy.stackexchange.com/a/33632/7982">this answer</a> 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 <strong>half of the period</strong> for the duration of the flight from Earth to Jupiter.</p></li>
<li><p>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 <a href="https://en.wikipedia.org/wiki/Elongation_(astronomy)" rel="nofollow noreferrer">read how elongation is defined</a>.</p></li>
</ol>

CC BY-SA 4.0
2019-10-22T13:07:46.430Z
2019-10-22T13:13:25.053Z
2019-10-22T13:13:25.053Z
uhoh
<p>Consider the Hohmann transfer orbit of a probe launched from Earth to Jupiter. Assume the orbits of Earth and Jupiter are circular.</p>

<p>I want to calculate the following:</p>

<ol>
<li>The semi-major axis and eccentricity of Hohmann transfer orbit from Earth to Jupiter.</li>
<li>The speed of the probe at Earth.</li>
<li>The launch speed of the probe from Earth to reach Jupiter.</li>
<li>The duration of the flight to Jupiter.</li>
<li>The elongation of Jupiter at launch to ensure that the probe arrives at Jupiter.</li>
</ol>

<p>Here my attempt:</p>

<ol>
<li><p><span class="math-container">$a = \frac{1}{2}(r_E + r_J) = \frac{1}{2}(1+ 5.203) = 3.102 \text{ AU}, \ \ e = 1 − \frac{r_E}{a} = 0.677$</span></p></li>
<li><p>I am thinking of using the vis-viva equation but I don't know what I should use for <span class="math-container">$r$</span>? Should I use <span class="math-container">$r_E$</span> = 1 AU for <span class="math-container">$r$</span> where <span class="math-container">$r_E$</span> is the average distance from the Earth to the Sun? Am I right? Please correct me if I am wrong.</p></li>
<li><p>I am thinking of using the escape velocity formula where <span class="math-container">$M$</span> is the mass of the Earth and <span class="math-container">$R$</span> is the radius of the Earth. Please correct me if I am wrong.</p></li>
<li><p><span class="math-container">$t = \frac{1}{2}P = \frac{1}{2} \sqrt{a^3} = 2.73 \text{ years}$</span>.</p></li>
<li><p>I really don't know how to do this question. Please help.</p></li>
</ol>

CC BY-SA 4.0
2019-10-22T12:45:10.113Z
It turns out that there are [46 questions tagged with `hohman-transfer`](https://space.stackexchange.com/questions/tagged/hohmann-transfer) in the related site Space Exploration Stack Exchange. I think you can find some helpful guidance there as well, but I'll leave a short answer for you now.
uhoh
CC BY-SA 4.0
2019-10-22T12:50:09.283Z
Thanks so much, much appreciated.
RUNN
<celestial-mechanics>
Hohmann transfer orbit (Earth-Jupiter system)
RUNN
# Hohmann transfer orbit (Earth-Jupiter system)

(Asked by: username_0 on 2019-10-22)

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.

**Tags:** celestial-mechanics

> **Comments:**
> **username_1**: It turns out that there are [46 questions tagged with `hohman-transfer`](https://space.stackexchange.com/questions/tagged/hohmann-transfer) in the related site Space Exploration Stack Exchange. I think you can find some helpful guidance there as well, but I'll leave a short answer for you now.  
> **username_0**: Thanks so much, much appreciated.  

---

## 1 answers

### Answer by username_1 on 2019-10-22. Score: 3

> 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](https://en.wikipedia.org/wiki/Hohmann_transfer_orbit) 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](https://en.wikipedia.org/wiki/Vis-viva_equation) whenever you can! The [standard gravitational parameter](https://en.wikipedia.org/wiki/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](https://astronomy.stackexchange.com/a/33632/7982) 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](https://en.wikipedia.org/wiki/Elongation_(astronomy)).


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