Calculating a planet\'s mass & gravitational pull
 

Help!

I'm putting together a spreadsheet to help me create realistic planets for SE4 story writing purposes.

I type in the diameter (km) and surface area covered by ocean (%), and it automatically gives me the planet's circumference (km), total surface area (km^2), land and water surface area(km^2), and compares all these values to Earth for easy reference.

What I'd like in addition is to be able to plug in a Mean Density value (gr./cm^3) for the planet and from that get an estimate of the planet's surface gravity.

Unfortunately I don't know the calculations necessary for this, despite lengthy google searches. Can anyone help?

I assume I'll need the planet's volume, but I should be able to get that from the diameter. I appreciate that any figures I come up with won't be utterly accurate (because I'm working with spheres and planets aren't quite spherical, and because surface gravity is also affected by rotation) but they'll be good enough for my purposes.

Re: Calculating a planet\'s mass & gravitational pull
 

Just multiply average density by volume.
kg/m^3 x m^3 = kg

Useful for rocky/icy planets, but will be trouble on gas giants due to the extreme changes in density going down to the core.

For surface gravity, F= GMm/r^2
Where gravitational constant G = 6.67300 × 10-11 m3 kg-1 s-2
Big and little masses M & m and the surface radius.
Little m will cancel out when you do F=ma to get the acceleration.
A = GM/R^2
Just remember to use the same units throughout.

PS:
Have fun picking a surface for your gas giant, and don't count any material above the surface in your mass calculation http://forum.shrapnelgames.com/images/icons/icon12.gif

[ August 20, 2003, 15:24: Message edited by: Suicide Junkie ]

Re: Calculating a planet\'s mass & gravitational pull
 

Thanks S_J- a few questions:
<font size="2" face="Verdana, Helvetica, sans-serif">So that's kilograms per metre cubed times metres cubed. My "volume" value is in kilometres cubed, so do I just divide by 1000 to get metres cubed?

<font size="2" face="Verdana, Helvetica, sans-serif">Glurp! so if my distant maths lessons are correct, that would be -6.67300 with the decimal place moved 11 spaces to the right. Correct? What does the kg-1 s-2 mean?

<font size="2" face="Verdana, Helvetica, sans-serif">so M in this one is the mass of the planet as calculated above.

If I run Earth's vital statistics through the above treadmill I should get a value somewhere near 1, is that right?

Re: Calculating a planet\'s mass & gravitational pull
 

1 cubic kilometer = 1,000,000,000 cubic meters (i.e. 1000^3 cubic meters)

Yes, 10^-11 means move the decimal 11 places to the left.

The kg-1 and s-2 refer to exponents in this limited UBB system. Negative means in the denominator, so the units would be: m^3 / [kg x s^2]

Slick.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Aha! So if I multiply the result by 1000000000... Yes, i get a value for Earth & Mars masses that (more or less) matches what I find on websites. Excellent, thanks, slick.

<font size="2" face="Verdana, Helvetica, sans-serif">OK, I'll plug that in.

<font size="2" face="Verdana, Helvetica, sans-serif">Ummm... I didn't understand any of that. Sorry, can you put it into language that can be understood by someone of my high mean density?

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Tomorrow I'm going to have a go at putting all of this functionality into java so I can offer it on my website.
Planned features:
-flexible input values (ie it can automatically calculate circumference from diameter or vice versa.)
-Side by side comparison with a selectable "benchmark" planet (Earth, Mars, Jupiter, Plenty... any others I should have?
-Choose between raw data comparison or proportional comparison (ie Planet X has 125% earth's land surface area)
-Variable input units (miles or kilometres etc)

Anyone likely to use this? I doubt it will see a lot of usage, but thought it might be a handy toy for sci-fi writers. Any particular features I ought to implement? Even if no-one wants it I'll do it anyway as a programming excercise for myself.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Actually, don't forget the stuff overhead when figuring gravity. It has mass, too. The gravity inside a gas giant could be very weird. The atmospheric pressure is probably a more predictable problem.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Aaron Hall? http://forum.shrapnelgames.com/images/icons/icon7.gif You ought to share it with him. It might improve SE V.

Re: Calculating a planet\'s mass & gravitational pull
 

The stuff above you will cancel out with the stuff at the same height on the other sides of the planet.
The stuff above your horizon pulls you away, while the stuff below your horizon at the same altitude pulls you towards (weaker due to distance, but more mass) Given reasonably constant density from one side to the other, it cancels out.

Stormy weather would make for small differences.
Sunlight heating one side would make for bigger differences.
But the density can't vary by that much, since the high pressure air will just blow into the low pressure zone and equalize it.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Ummm... I didn't understand any of that. Sorry, can you put it into language that can be understood by someone of my high mean density?

--------------------------------------------

I appologize for being hard to understand. The limitations of this UBB make it hard to use normal mathematical notation.

The units of the gravitational constant G are:

</font><blockquote><font size="1" face="Verdana, Helvetica, sans-serif">code:</font><hr /><pre style="font-size:x-small; font-family: monospace;"> 3
m
------------------
2
(kg) (s) </pre><hr /></blockquote><font size="2" face="Verdana, Helvetica, sans-serif">Hope this helps.

Slick.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Not quite. It is not negative, just really close to 0. http://forum.shrapnelgames.com/images/icons/icon12.gif A negative exponent does not change the sign of the base number.

Re: Calculating a planet\'s mass & gravitational pull
 

right, so 6.67300 × 10-11 m3 kg-1 s-2 could be written as:

6.673 x 10^-11 m^3/kg*s^2

or

0.00000000006673 m^3/kg*s^2

I like to put the breaks in every 1000 so I usually write this:

0.00 000 000 006 673

This, as IF said, is quite different than:

-0.00 000 000 006 673

I hope that this is helpful, and isn't construde as beating a dead horse

[ August 21, 2003, 04:31: Message edited by: jimbob ]

Re: Calculating a planet\'s mass & gravitational pull
 

Thanks everyone, I think I have the spreadsheet working now. For Earth I get a gravity value of 9840866.19 (starting with a diameter of 12756km and density of 5520kg/m^3). For Mars I get 3736591.2 (diameter 6794.4, density 3935). That's exactly right in proportion to one another, but I have no idea whether or not the values themselves are accurate- all the websites that list such data measure surface gravity in comparison to Earth's.
Also, I haven't factored in rotation which (apparently) increases the effective gravitational pull.

I'm still a little fuzzy on the kg-1 s-2, although I now understand that it's just the unit by which gravity is measured. How would you pronounce it? What does that s stand for? Is it seconds?

=-=-=-=-==-=-=-=-=-=-==-=-=-

Any O&C fans in the audience (if I haven't killed them all off with the lack of updates) may be interested to know that I've now used the spreadsheet to come up with some values for Ceres IV (Plenty).
diameter: ~19000km.
Circumference: ~60,000 km.
Surface area: 2.2 times that of Earth, but I want at least 85 or 90% of it to be water, which leaves betwee 1.15 and 0.77 times Earth's land area. I'll probably go for 90%.
Density: Not sure yet. With an Earth-like density, it would be about 1.5 times Earth gravity (bloody hard work, but probably livable). Interestingly enough, with a Mars-like density, the number comes out only a little above Earth's gravity.

I always intended it to be high in mins and orgs, but very low in rads, so that would fit in well with low density and therefore only-slightly-higher-than-Earth gravity, and it would also help to explain (along with all the beaches) why the place is so damned popular.

Of course, all these numbers are muddied in the text by literary vagueness, so none of them have to be particularly precise.

Re: Calculating a planet\'s mass & gravitational pull
 

<font size="2" face="Verdana, Helvetica, sans-serif">Did you remember convert the diameter to meters? The answer (9840866.19) is right, but the unit would be μm/s^2.

<font size="2" face="Verdana, Helvetica, sans-serif">Yes s is seconds. s-2 is equal to s^-2 which is equal to 1/s^2

Re: Calculating a planet\'s mass & gravitational pull
 

Hmm... I hadn't done that. However if I divide the diameter by 1000 to get metres, my gravity figure comes out even bigger. Or is that right?

Re: Calculating a planet\'s mass & gravitational pull
 

Let's see what I remember from my physics classes.
Gravity on earht should be 9.8 m/s2 pronounced "meters per square second". Meaning the speed of an object in free fall will increase in 9,8 m/s every second.
So you seem to have the point shifted several spaces.

No, rotation should actually create a centrifugal force that will pull you away from the planet and tend to lower gravity close to the equator.

Re: Calculating a planet\'s mass & gravitational pull
 

You need to multiply by 1000 instead of divide, since there are more meters than kilometers.

Re: Calculating a planet\'s mass & gravitational pull
 

DOH! That's pretty much how NASA missed Mars with the Last mission. Or maybe they hit Mars, but they certainly didn't get into orbit as desired. It was a unit conVersion error. http://forum.shrapnelgames.com/images/icons/icon12.gif

Slick.

Re: Calculating a planet\'s mass & gravitational pull
 

It helps if you set up "conVersion factors" (lame, yes; but still useful). An example:

</font><blockquote><font size="1" face="Verdana, Helvetica, sans-serif">code:</font><hr /><pre style="font-size:x-small; font-family: monospace;"> / 1000 m \
12 km x | ------ | = 12000 m
\ 1 km / </pre><hr /></blockquote><font size="2" face="Verdana, Helvetica, sans-serif">(those are parenthesis)

Re: Calculating a planet\'s mass & gravitational pull
 

Yeah, one of my planned features will be the ability to select different units for input/output. I'll just have to filter all calculations through a "conVersion" function.

Re: Calculating a planet\'s mass & gravitational pull
 

Why not just output it in all units simultaneously.
B5 = meters
B7 = B5 * feet/meter
...etc...

Re: Calculating a planet\'s mass & gravitational pull
 

That would make it too cluttered. I'm not talking about the spreadsheet any more, I'm currently putting the whole thing into javascript for my webpage.

Re: Calculating a planet\'s mass & gravitational pull
 

Looks good to me

Re: Calculating a planet\'s mass & gravitational pull
 

Yeah, I thought it looked ok, but all the numbers come out wrong.
Here's the line from the code. Something just has to be wrong with it, but I'll be stuffed if I can figure it out.

cells[radius].value=Math.pow(((cells[gravity].value/GRAV)/cells[mass].value),1/3)/1000;

I've also just put in an alert statement to display the output of the same calculation (in case the number was getting jiggered about elsewhere) but apparently it isn't.

[ August 27, 2003, 14:11: Message edited by: dogscoff ]

Re: Calculating a planet\'s mass & gravitational pull
 

Hmm, I was *way* off, but now I'm closer. I went back to S_J's original calculation of g = GM/(1000*R)^2
(dunno where I got g=G*m*(r*1000)^3 from)

g = GM/(1000*R)^2
g*(1000*R)^2 = GM
(1000*R)^2 = GM/g
(1000*R) = squareroot(GM/g)
R=(squareroot(GM/g))/1000

In javascript, that comes out as
radius=(Math.sqrt(GRAV*mass)/gravity)/1000

and it works. Sorry for all the fuss, ppl.

[ August 27, 2003, 15:12: Message edited by: dogscoff ]

Re: Calculating a planet\'s mass & gravitational pull
 

radius=(Math.sqrt(GRAV*mass)/gravity)/1000

Shouldn't you Math.sqrt( grav*mass/gravity ) or somesuch?

Re: Calculating a planet\'s mass & gravitational pull
 

GURPS Space (from Steve Jackson Games) used to have work sheets for generating alien planets, that included orbital radius and period, roation, gravity, mass, temerature, etc, etc..

might look for an old copy, or a PDF of the reference sheets Online.

Re: Calculating a planet\'s mass & gravitational pull
 

Hmm... I've often wondered about the error associated with using point source gravitational equations. Too lazy to look it up, so maybe someone here knows.

The equations posted here all assume that gravity comes from a point source at the core of a planet. This is fine when you are many, many planet diameters away from a planet. But when you get within some distance, the gravitation effects should be distributed across the planet's height, width, and thickness. The gravitational force that a body feels when within this distance is the vectored sum of all these components. Additionally, each layer of a planet has a different density and therefore differing gravitational contributions.

A second option is that during planet formation, the denser elements settle more towards the planet core. In effect, the bulk of the planet's mass resides in the core. This would make the point source error very small until you've pierced the surface and moved very close to the planet's core.

Re: Calculating a planet\'s mass & gravitational pull
 

*sigh*

sorry to bother y'all again, but...
Where am I going wrong? I'm trying to work out the calculation needed to derive radius (in km) from a known mass and surface gravity (in metres per second per wotsit):

g=surface gravity, G= gravitational constant, m=mass, r=radius.

g=G*m*(r*1000)^3 - this one works in my excel sheet.
therefore
g/G=m*(r*1000)^3
therefore
(g/G)/m=(r*1000)^3
therefore
cuberoot((g/G)/m)=r*1000
therefore
cuberoot((g/G)/m)/1000=r

Or is that all crud? It's a long time since I studied or used this kind of maths.

Apart from this, the javascript sheet is coming along nicely.

Re: Calculating a planet\'s mass & gravitational pull
 

Ack: I don't doubt that these equations have a certain level of innaccuracy built in, but I'm only using them for writing purposes, so I can be sufficiently vague to leave room for correction.
When the program goes up on my website though I fully intend to list all the known areas of innaccuracy. I'll add your comments to my list, thanks=-)

Re: Calculating a planet\'s mass & gravitational pull
 

Differing densities at various depths does not affect the calculation, assuming that at any particular depth, the density is the same all the way around the planet.
If the density is not the same (ocean water vs rock, say) it still only has a small effect, which is swamped by the effect of the 1000's of km of rock below it.

Distance from the planet surface has the biggest effect.
If you are on top of a mountain, the gravity is lower. While there is a bit more mass directly below you, the 1/r² decrease hits much harder.

However, its still a fraction of a percent change.

You'd need something with an irregular shape, like an asteroid to get decent changes in gravity, but then the gravity is so low to begin with, it really dosen't matter.

Stuff with higher gravity smushes itself into an ovoid ball, and the approximations apply.

Re: Calculating a planet\'s mass & gravitational pull
 

Wahey! It's finished. Well, not quite finished, but the difficult bit is.

You can put in any combination of variables and it will work out everything it can from them. For example, you could put in radius and mean density to get surface gravity, mass, volume, circumference etc.

Equally though, you could put in surface gravity, circumference and percentage of the surface covered in water and from that derive the radius, mean density and total land area.

On the other hand, you could just tell it how much land and ocean surface there is and the surface gravity and it will give you density, volume, etc etc etc...

You get the idea. If anyone ever needs to design a planet to spec then this could be handy.

All that remains is to validate the input a little more, make the option to compare the figures generated to those of real planets, pretty it up a bit and then I can upload it to my website for everyone to ignore http://forum.shrapnelgames.com/images/icons/icon10.gif

Thanks for all the help, ppl.