<p>Compare the orbital speeds and use the superposition.</p>
<p>Jupiter's orbital speed around the sun is around 13km/s. Io's speed around Jupiter is around 17km/s. So it will actually go backwards.</p>
<p>For plotting also apply the superposition principle of the motions.</p>
<p>How to create such plot: Assuming that the moon has a circular orbit with period T_m, you can describe its motion in cartesian coordinates as <span class="math-container">$x_m(t) = r_m\cdot\sin(2\pi t/T_m) $</span> and  <span class="math-container">$y_m(t) = r_m\cdot\cos(2\pi t/T_m) $</span>. Add to that the movement of Jupiter with its orbital period of <span class="math-container">$T_J = 12$</span>years and distance <span class="math-container">$r_J$</span> around the sun: <span class="math-container">$x_J(t) = r_J\cdot\sin(2\pi t/T_J) $</span> and <span class="math-container">$x_J(t) = r_J\cdot\cos(2\pi t/T_J) $</span>. Calculate x and y for small time steps (probably hours or even minutes, given the moon's small orbital period of a few days), but cover a complete orbit around jupiter and plot the resulting positions.</p>
<p>I cannot plot right now with my phone. But @MikeG meanwhile <a href="https://www.desmos.com/calculator/ocwbchetyp?lang=de" rel="nofollow noreferrer">plotted it:</a><a href="https://i.sstatic.net/cIYL0.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cIYL0.png" alt="Plot of the Galilean Moons by @MikeG" /></a>
where the black dashed line is Jupiter's orbit around the Sun, and orange, blue, green and red are the orbits of Io, Europa Ganymede and Calisto respectively using <a href="https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html" rel="nofollow noreferrer">these data</a>.</p>
<p>Additionally, <a href="https://www.desmos.com/calculator/uow4ojd0t1" rel="nofollow noreferrer">this Desmos plot</a> shows the orbits of all large round moons around their respective planet, assuming circular orbits. Note that some satellites' orbits (namely Enceladus and Dione) might appear janky; zooming out or in might fix the issue.</p>
<p>In summary: Io, Europa, Mimas, Enceladus, Tethys, and Dione are the only large round moons that have “loops” in their orbits. All other large satellites have no retrograde part in their orbit with respect to the Sun, including Ganymede and Calisto as shown in the plot above.
<a href="https://en.wikipedia.org/wiki/Convex_curve" rel="nofollow noreferrer">Convex</a> is a stricter criterion than &quot;not <a href="https://en.wikipedia.org/wiki/Apparent_retrograde_motion" rel="nofollow noreferrer">retrograde</a>&quot; and geometrically means that I can take any arbitrary point on the orbit and reach on a straight line any other point of the orbit and remain within the area encircled by the orbit. Mathematically it means that the sign of the curvature of the orbit around the Sun does not change. Looking at the moons' orbits, the geometrical explanation of concave makes it immediately clear that none of them is a concave orbit.</p>

CC BY-SA 4.0
2023-11-24T16:24:58.613Z
[Plotted in Desmos](https://www.desmos.com/calculator/ocwbchetyp) using [these data](https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html) (units: day, million km)
Mike G
CC BY-SA 4.0
2023-11-24T18:15:02.327Z
@MikeG this is exactly what I was looking for, thank you. If you'll add it to the answer you replied to, I can accept it.
user267545
CC BY-SA 4.0
2023-11-24T18:38:56.707Z
@MikeG that's an awesome plot. With your assumed permission I added it here (please revert, if you disagree or want to convert that into your own answer)
planetmaker
CC BY-SA 4.0
2023-11-24T22:24:34.293Z
@planetmaker You're welcome to use it.
Mike G
CC BY-SA 4.0
2023-11-25T01:59:37.033Z
@MikeG any chance the lines can be thicker or darker for the purposes of this post? https://i.sstatic.net/ZybMg.jpg
uhoh
CC BY-SA 4.0
2023-11-25T04:30:41.757Z
@uhoh Not my screenshot. Desmos lets you change the line width; follow the link and look for a gear widget.
Mike G
CC BY-SA 4.0
2023-11-25T05:43:02.397Z
@MikeG nice! :-)
uhoh
CC BY-SA 4.0
2023-11-25T09:52:50.520Z
@user267545 convex is more strict than "not retrograde". All orbits shown in this plot are both, concave and convex, depending on which part of the orbit you look at. Concave means there is no part of the sign of curvature changes, but it always "bends" towards the sun, more or less, thus either going straight or turning left, manner-of-speaking.
planetmaker
CC BY-SA 4.0
2023-11-25T09:56:02.527Z
@MikeG thanks for the hint with the settings. I didn't find nor see those when I took the screenshot.
planetmaker
CC BY-SA 4.0
2023-11-25T16:26:47.177Z
@planetmaker TIL Desmos lets you put equations in folders, thanks to your example.
Mike G
CC BY-SA 4.0
2023-11-25T17:00:59.243Z
@MikeG I did not waste an hour and a half of my life plotting each moon and planet's orbital characteristics into Desmos for someone else to take the credit :p That said I just copied the format that you made for the Galilean moons so ultimately it's all thanks to you.
user267545
CC BY-SA 4.0
2023-11-24T00:31:55.850Z
2023-11-25T10:35:23.267Z
2023-11-25T10:35:23.267Z
planetmaker
<p>The shape of the Moon's orbit around the Sun <a href="https://physics.stackexchange.com/questions/266426/what-does-the-moons-orbit-around-the-sun-look-like">has been discussed</a> to death, and the answer is that the Moon's orbit has no concave &quot;loops&quot; whatsoever.</p>
<p>I am relatively sure that the determinant factor in whether a moon has a &quot;concave&quot; orbit with loops or a loop-less orbit has to do with its orbital velocity. If the moon orbits its planet faster than the planet orbits the Sun, then the moon has the intuitively-expected looped orbit. If not, then it is loopless.</p>
<p>From my calculations, Io and Europa orbit Jupiter faster than Jupiter orbits the Sun, while Ganymede and Callisto orbit slower than Jupiter orbits the Sun. Therefore, it's expected that Io and Europa make loops around Jupiter while it is orbiting the Sun, while Ganymede and Callisto don't.</p>
<p>This is all great, except I have no clue how to verify this hypothesis. I originally thought the only way to see was to run a simulation, however I'm 90% sure it can be done by graphing parametric equations using WolframAlpha or Desmos.</p>
<p>One problem: I have exactly zero idea how to go on doing that, and barely know parametric equations exist to begin with. So if someone smarter than me could graph the Galilean moons' orbits and verify if Io and Europa have loops while Ganymede and Callisto don't, I'd appreciate it.</p>

CC BY-SA 4.0
2023-11-24T00:27:43.507Z
Take a look at my orbit plotter at the end of this answer: https://astronomy.stackexchange.com/a/49823/16685
PM 2Ring
CC BY-SA 4.0
2023-11-25T01:55:29.680Z
It's been discussed "to death" right here in Astronomy SE as well! https://astronomy.stackexchange.com/a/35408/7982
uhoh
<orbit><orbital-mechanics>
Do the Galilean moons' orbits around the Sun have loops (i.e. have concave parts)? And if so, how can their orbits be graphed?
user267545
# Do the Galilean moons' orbits around the Sun have loops (i.e. have concave parts)? And if so, how can their orbits be graphed?

(Asked by: username_0 on 2023-11-23)

The shape of the Moon's orbit around the Sun [has been discussed](https://physics.stackexchange.com/questions/266426/what-does-the-moons-orbit-around-the-sun-look-like) to death, and the answer is that the Moon's orbit has no concave "loops" whatsoever.

I am relatively sure that the determinant factor in whether a moon has a "concave" orbit with loops or a loop-less orbit has to do with its orbital velocity. If the moon orbits its planet faster than the planet orbits the Sun, then the moon has the intuitively-expected looped orbit. If not, then it is loopless.

From my calculations, Io and Europa orbit Jupiter faster than Jupiter orbits the Sun, while Ganymede and Callisto orbit slower than Jupiter orbits the Sun. Therefore, it's expected that Io and Europa make loops around Jupiter while it is orbiting the Sun, while Ganymede and Callisto don't.

This is all great, except I have no clue how to verify this hypothesis. I originally thought the only way to see was to run a simulation, however I'm 90% sure it can be done by graphing parametric equations using WolframAlpha or Desmos.

One problem: I have exactly zero idea how to go on doing that, and barely know parametric equations exist to begin with. So if someone smarter than me could graph the Galilean moons' orbits and verify if Io and Europa have loops while Ganymede and Callisto don't, I'd appreciate it.

**Tags:** orbit, orbital-mechanics
> **Comments:**
> **username_1**: Take a look at my orbit plotter at the end of this answer: https://astronomy.stackexchange.com/a/49823/16685  
> **username_2**: It's been discussed "to death" right here in Astronomy SE as well! https://astronomy.stackexchange.com/a/35408/7982  

---

## 1 answers

### Answer by username_3 on 2023-11-24. Score: 5 | Accepted answer

Compare the orbital speeds and use the superposition.

Jupiter's orbital speed around the sun is around 13km/s. Io's speed around Jupiter is around 17km/s. So it will actually go backwards.

For plotting also apply the superposition principle of the motions.

How to create such plot: Assuming that the moon has a circular orbit with period T\_m, you can describe its motion in cartesian coordinates as $x\_m(t) = r\_m\cdot\sin(2\pi t/T\_m) $ and $y\_m(t) = r\_m\cdot\cos(2\pi t/T\_m) $. Add to that the movement of Jupiter with its orbital period of $T\_J = 12$years and distance $r\_J$ around the sun: $x\_J(t) = r\_J\cdot\sin(2\pi t/T\_J) $ and $x\_J(t) = r\_J\cdot\cos(2\pi t/T\_J) $. Calculate x and y for small time steps (probably hours or even minutes, given the moon's small orbital period of a few days), but cover a complete orbit around jupiter and plot the resulting positions.

I cannot plot right now with my phone. But @MikeG meanwhile [plotted it:](https://www.desmos.com/calculator/ocwbchetyp?lang=de)[![Plot of the Galilean Moons by @MikeG](https://i.sstatic.net/cIYL0.png)](https://i.sstatic.net/cIYL0.png)
where the black dashed line is Jupiter's orbit around the Sun, and orange, blue, green and red are the orbits of Io, Europa Ganymede and Calisto respectively using [these data](https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html).

Additionally, [this Desmos plot](https://www.desmos.com/calculator/uow4ojd0t1) shows the orbits of all large round moons around their respective planet, assuming circular orbits. Note that some satellites' orbits (namely Enceladus and Dione) might appear janky; zooming out or in might fix the issue.

In summary: Io, Europa, Mimas, Enceladus, Tethys, and Dione are the only large round moons that have “loops” in their orbits. All other large satellites have no retrograde part in their orbit with respect to the Sun, including Ganymede and Calisto as shown in the plot above.
[Convex](https://en.wikipedia.org/wiki/Convex_curve) is a stricter criterion than "not [retrograde](https://en.wikipedia.org/wiki/Apparent_retrograde_motion)" and geometrically means that I can take any arbitrary point on the orbit and reach on a straight line any other point of the orbit and remain within the area encircled by the orbit. Mathematically it means that the sign of the curvature of the orbit around the Sun does not change. Looking at the moons' orbits, the geometrical explanation of concave makes it immediately clear that none of them is a concave orbit.

> **Comments:**
> **username_4**: [Plotted in Desmos](https://www.desmos.com/calculator/ocwbchetyp) using [these data](https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html) (units: day, million km)  
> **username_0**: @MikeG this is exactly what I was looking for, thank you. If you'll add it to the answer you replied to, I can accept it.  
> **username_3**: @MikeG that's an awesome plot. With your assumed permission I added it here (please revert, if you disagree or want to convert that into your own answer)  
> **username_4**: @planetmaker You're welcome to use it.  
> **username_2**: @MikeG any chance the lines can be thicker or darker for the purposes of this post? https://i.sstatic.net/ZybMg.jpg  
> **username_4**: @uhoh Not my screenshot. Desmos lets you change the line width; follow the link and look for a gear widget.  
> **username_2**: @MikeG nice! :-)  
> **username_3**: @user267545 convex is more strict than "not retrograde". All orbits shown in this plot are both, concave and convex, depending on which part of the orbit you look at. Concave means there is no part of the sign of curvature changes, but it always "bends" towards the sun, more or less, thus either going straight or turning left, manner-of-speaking.  
> **username_3**: @MikeG thanks for the hint with the settings. I didn't find nor see those when I took the screenshot.  
> **username_4**: @planetmaker TIL Desmos lets you put equations in folders, thanks to your example.  
> **username_0**: @MikeG I did not waste an hour and a half of my life plotting each moon and planet's orbital characteristics into Desmos for someone else to take the credit :p That said I just copied the format that you made for the Galilean moons so ultimately it's all thanks to you.  

---

