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I know this is more of a science question but my Google-fu isn't quite answering the question to my needs and it really is game related, I swear.
So I'm whipping up a new game setting that is based sort of on the Galilean moons of Jupiter. It's four small worlds orbiting a gas giant planet. The worlds are smaller than a typical Earth-sized fantasy setting with two that are roughly the same surface area as our moon and two about twice the surface area, about the size of Mercury. So, basically, worlds similar to the Galilean moons.
Anyway, I am trying to determine how big the planet will appear in the sky over these worlds. I've done a fair amount of research and gotten lots of data on the subject, but I am failing at translating that to something simple I can relate to my players. So some help from you fine, smart folks would be much appreciated.
The unit of measurement I am using both out of game and in game is a yardstick. If the ruler is held at arm's length how big does the object appear in the sky? I checked and Earth's moon looks to be roughly 3/4 of an inch in diameter. Since my worlds all orbit their host planet at roughly the same distance as the Galilean moons, how big would Jupiter appear from the surface of each?
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Diameter of jupiter / radius of orbit = angle from moon.
I used 86,881mi for diameter and 262,000mi for orbital distance (yuck miles).
I get .33 radians or 19 degrees. Wow pretty huge, someone should double check that.
We'll say the average human arm is 25 inches.
That means Jupiter measures 8 and a quarter inches with a yardstick at arms length.
I doubled checked this for our moon and got 30 arc minutes which the internet agrees with. I guess this is quazi accurate.
Edit:Whoops. The angle calculation is correct but the ruler one is not. I'll try to fix it later but I must be off!!
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"Big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts..."
Huh I think my ruler calculation is right. However, my calculation says our moon is 1/4 of an inch.
Regardless, as far as angular size goes Jupiter would be more than 15 times larger than the moon in the sky by diameter. Enormous. They could see great detail in it's storms and clouds.
Here is a youtube video imagining different planets in the moon's orbit around earth. Io and the moon have similar orbital distances, so this would be comparable.
Kazaan your numbers seem enormous to me. How are you getting arc minutes?
(object diameter/orbital distance)*3348 (3348 is radian to arc minute).
Using that formula I find it to be muuuuuch smaller than the values you posted.
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Thanks guys. I understand the physics behind perspective and such but when searching online I kept getting measurements in degrees and it wasn't computing in my head and certainly wasn't anything I could translate to my ruler system. The other thing I was getting were artist renderings from the surface of the moons and that was nice to give a sense of perspective, but still didn't tell me how big the planet would appear.
These measurements don't need to be precise, just something ballpark I can tell my players. I want to hand them a ruler and make them go to a window and do this so they can get an idea of the scale. I can't just tell them that from the closest world the planet looks REALLY big and from the farthest world it still is pretty big. The best I have so far is a simple comparison of Jupiter from Io appearing 36x bigger than the moon from Earth which would make it almost the size of the entire yardstick and that seems excessive.
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This should help:
https://www.youtube.com/watch?v=usYC_Z36rHw
As Donald Trump once said,
For reference, the angular size of the full moon from Earth is 31 minutes of arc. (or, 0.5°).
According to Wikipedia...
Jupiter has a mean diameter of about 140,000 km.
The orbital radii (i.e. distance to the center of Jupiter) of the Galilean moons are:
Io: 422,000 km
Europa: 671,000 km
Ganymede: 1,070,000 km
Callisto: 1,880,000 km
Now, using this online angular size calculator, plugging in these values gives the angular size of Jupiter from...
Io: 1130" (19°; about 40x size of full moon from Earth)
Eurpoa: 715" (or, 13°; about 25x size of full moon)
Ganymede: 449" (or, 7.5°; about 15x size of full moon)
Callisto: 256" (or, 4.5°; about 10x size of full moon)
Jupiter will clearly dominate the skies of any of its moons. But, how do those values translate?
1 degree of arc in the sky is about the width of your little finger at arm's length. If you go out tonight and hold out your arm, you'll see that the moon is about half that size.
Your outstretched fist, held vertically, spans about 5°; that's the apparent size of Jupiter from Callisto.
Your outstretched fist held horizontally spans about 10°. That's only a little larger than the apparent size of Jupiter from Europa.
If you make the "Rock!" gesture (i.e. ASL "Y"), the span from your index finger to your little finger spans about 15°. That's a little larger than the apparent size of Jupiter from Europa.
And, if you make the "Longhorns" gesture, the span from your thumb to your little finger is about 20°. That's about the size of Jupiter from Io.
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Okay, let's see if I can make these equations work. These numbers are super rough to make the math easier because I don't need super specific.
According to the internet the moon from earth looks to be about the size of a dime held at arm's length. Also according to the internet the diameter of a dime is just over .7 inches so roughly the measurement I got with my incredibly precises method of standing on my porch when the moon was full the other day, holding up my fingers, and guessing.
Here goes:
The diameter of Jupiter is 140,000km.
Io is 422,000km from Jupiter.
Europa is 671,000km from Jupiter.
Ganymede is 1,070,000km from Jupiter.
Callisto is 1,883,000km from Jupiter.
Applying the formulas above and assuming I did the math correctly (I rounded way off), I get the following values:
Io: .33 radians or 1,105 arc minutes
Europa: .21 radians or 703 arc minutes
Ganymede: .13 radians or 435 arc minutes
Callisto: .07 radians or 234 arc minutes
My math seems to agree with the above calculation of the moon and Earth being .009 radians and 30 arc minutes. So all this tells me that 30 arc minutes equals 0.7 inches (the diameter of a dime). So that creates the following formula: (arc minutes/30)*0.7
Applying all of this math gives me the following values of how big Jupiter will appear on a yardstick from arm's length:
Io: 25.76 in
Europa: 16.38 in
Ganymede: 10.15 in
Callisto: 5.46 in
All told, really damn big. Thanks for the help in getting these numbers guys, I appreciate it. Any double checking of my math would be greatly appreciated.
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EDIT: Thanks Haladir, it looks like we came to different results as the numbers got bigger.
Go watch the movie The Europa Report. It's about a manned space flight to Europoa which IS one of Jupiter's Galilean moons. The special effects are pretty good and there are several SFX views of Jupiter seen from Europa's surface.
It's not real, of course. Just CGI, but apparently they had a few actual scientists who consulted on the movie to get stuff like this right so it's probably accurate enough for your roleplaying game.
If you don't have time or access to the whole movie, here's a bit of a clip that I found n youtube: it starts at 0:40.
To be clear, I'm not saying it's a great movie. I enjoyed it; it was fun, but I won't call it great. My only point was that the CGI scenes with Jupiter on the horizon are probably exactly what you're looking for.
1 degree of arc in the sky is about the width of your little finger at arm's length. If you go out tonight and hold out your arm, you'll see that the moon is about half that size.Your outstretched fist, held vertically, spans about 5°; that's the apparent size of Jupiter from Callisto.
Your outstretched fist held horizontally spans about 10°. That's only a little larger than the apparent size of Jupiter from Europa.
If you make the "Rock!" gesture (i.e. ASL "Y"), the span from your index finger to your little finger spans about 15°. That's a little larger than the apparent size of Jupiter from Europa.
And, if you make the "Longhorns" gesture, the span from your thumb to your little finger is about 20°. That's about the size of Jupiter from Io.
This is great Haladir. Very useful for gaming purposes. The DM can set the picture easily with the "measurements" you described. I questioned the longhorns equally about 20° until a started placing them next to each other at arms lengths and by 5 I had passed over my own head going more than a quarter revolution and confirmed it enough for my requirements. Establishing the number at something slightly greater than 18.
I just did a Google image search and found some pretty good artists' conceptual illustrations.