User blog:Cerne/Lagrange Points

Hello there, now for another blog entry. It is after midnight and I am using a personal laptop so I will try to keep my part of the entry brief.

One thing I think I forgot to say in my last entry was that one reason why I am going to put certain pieces of information in it, like all that stuff I put in my last entry about determining rotation velocity and rotation period length, as well as snippets of thread posts (and perhaps emails too), is because these sources of information can not be found anywhere else. If I am going to use references or redirect people to the places where I found or came up with the information present in one of my articles, I will need to send them to a permanent source for the information, and unfortunately many of the sources for the information that I will be using are either temporary - and hence are already gone or will be soon - or have no concrete presence on the internet.

In the case with my data on determining my planet's rotation speed and day length, I wanted to explain how I did it so that people would know I determined it somehow and didn't just make it up on the spot. But I didn't want to explain it every single time someone asked, so instead I stored everything I did to come up with my results here. In this way, the data acts like an official or unofficial source of information instead of being merely an educated guess, and I can use it as a reference. This will be my own information so of course I have permission. You may verify it all you want, and if you do find that it came from somewhere or someone other than myself I assume full responsibility. But I reserve the right to ask for an explanation of why you think I took the information from another source and how I could have done so.

In the case with the information I am about to copy and paste from a saved online bboard thread, it is more about providing proof that the information was obtained from a real outside reference other than myself but that no longer exists anymore. In which case I will provide the person's online user name and where you may contact them to verify the information being displayed as well as where you can obtain more information on your own (at the discretion of the person you are contacting). Email addresses and other direct means of contact will be omitted except when permission by said person or people has been obtained. If any first, second or third party objects to what is being displayed in this or any other blog post, please leave a comment on the entry in question and I will make it my personal responsibility to amend the problem as best I can in a way that will be satisfactory to you. And I assume full responsibility if a comment was not dealt with to your satisfaction as long as you can show that you did leave a comment and/or the existing comment in contention was left by you.

As it has to do with emails - though I doubt it will - the email's sender will be notified and permission will be granted prior to posting the email here. I really doubt there will be emails posted in this blog since none of the people I email have sent me anything I would want to post here or that I would think was relevant to post here, but I felt I had to bring up emails since it is possible to post emails in a blog like this one.

And as for external sources of information on and off the internet, I will provide links to these sources in the article itself so there will be no need for it to show up in my blog.

Now, without any further ado, here are some equations that a member of the Zompist Bulletin Board (ZBB) posted in a thread I made about Lagrange Points. The equations themselves are more of a side product since they never directly addressed the question being asked in the thread's original post, but they were very helpfull nontheless. The original question had already been dealt with by that time, anyway.

Mars, Phobos, and Deimos, would be a good set.

Mars's mass is about 6.4185 * 10^23 kg, or about 10.7% of Earth's.

Phobos's mass is about 1.072 * 10^16 kg, or about "1.8 nano-Earths" Deimos's mass is about 1.48 * 10^15 kg.

Mars is about 60,000,000 times as massive as Phobos, and about 430,000,000 times as massive as Deimos. Phobos is about 1.7*10^-8 of the mass of Mars, and about 7.2 times as massive as Deimos. Deimos is about 2.3*10-9 the mass of Mars, and about 0.14 times the mass of Phobos.

Phobos's average distance from Mars is about 9,377.2 km, about 0.4 times as far away as Deimos. Deimos's average distance from Mars is about 23,460 km, about 2.5 times as far away as Phobos.

I have read that if the distance between the closest pair is at most one-fifth the distance between the most-widely-separated pair, one can usually treat their orbits as independent of each other.

That probably depends some on mass and distance, though.

We can work that out.

Suppose the planet's mass is M, the further moon's mass is M, and the nearer moon's mass is m.

Suppose the distance from the planet to the further moon is R, and the distance from the planet to the nearer moon is r.

The gravitational force between the planet and the further moon is proportional to (M*M)/(R^2).

The gravitational force between the planet and the nearer moon is proportional to (M*m)/(r^2).

The gravitational force between the further moon and the nearer moon can never be bigger than something proportional to (M*m)/((R-r)^2).

We'd like to make sure that the gravitational force between the moons is always less than 1% of the gravitational force between the planet and either moon.

If M is at least 102 times m and M is at least as massive as m but less massive than M, and R is at least 102 times r, then that will work.

That is a sufficient condition; but it's obviously not a necessary one.

For instance, if M is 10,000 times M and m is 10 times M, and R is between 10 and 10,000 times r, it will work out.

If

M is the ratio of the planet's mass to the mass of the less massive satellite, and

m is the ratio of the mass of the more massive satellite to the mass of the less massive satellite, and

R is the ratio of the orbital radius of the more distant satellite to that of the closer satellite,

so that

M > m >= 1

and

R > 1,

the force between the two satellites is never greater than something proportional to m/((R-1)^2).

If the nearer satellite is also the more massive satellite, the force between the planet and the nearer, heavier satellite is proportional to (M*m) and the force between the planet and the further, lighter satellite is proportional to M/(R^2).

If the further satellite is also the more massive satellite, the force between the planet and the nearer, lighter satellite is proportional to&nbsp M and the force between the planet and the further, heavier satellite is proportional to (M*m)/(R^2).

The force between the two satellites will be less than 1% of the other two forces, however they are arranged, provide

M > (100*m*(R^2))/((R-1)^2)

And, if M > 100*m, the force between the two satellites will be less than 1% of the other two forces, however they are arranged, provide

R > (M + 10*SQRT(M*m))/(M-100*m).

The more the planet outmasses the heavier satellite, the less the distance-ratio R is constrained to be high. The higher the distance-ratio R is, the less the mass-ratio M/m is constrained to be high.

If R>202 then for M >= 101*m things will work out;

if M > 401*m then for R >= 2 things will work out.

Now, we haven't said anything about stability. Presumably both satellites will have orbited the planet many millions of times, and will continue to orbit it for many millions more orbits. Over such a long time-scale, that force that's less than 1%, might in fact have an influence. There are certain ratios of the satellites' orbital periods that are markedly stable, and certain others that are markedly unstable. One of the stable ratios is 2:1. (edit:) Looks like that might be wrong; it seems the 2:1 ratio is usually unstable, though sometimes it is stable. It seems to depend on the masses and certain other orbital parameters. (/edit)

The length of time an orbit takes is proportional to the 3/2 power of the radius. So the more distant satellite's orbital period will be

R^(3/2) = T times the length of the nearer satellite's period. To have T = 2, then, you need R = 2^(2/3) which is about 1.587401052.

The above information was graciously provided in two posts, split here by line breaks, by TomHChappell from the ZBB (http://www.spinnoff.com/zbb). You may also reach him here at the Conworld Wikia through his user page: http://conworld.wikia.com/wiki/User:Eldin_raigmore

The rest of the information in the thread was also very useful but I would not need to post it here since I will not be using it for my conworld. Part of the purpose for posting information from online board threads is so that I may save or free up space in my email inbox. Whenever a thread is about to be pruned from the ZBB - other boards/forums I visit don't delete their threads - and I think it may be of use to me, I save it as an attachment in my inbox. Hopefully I may not need to keep doing this too much longer. Problem is, once I save the thread as an email attachment, I am unable to upload the file back onto a computer or onto disk. So once it is already saved, I can only cut and paste the text inside it. I would make a habit of saving onto disk instead but thread survival is awefully precarious on boards that prune their threads so my safest bet is to save the thread in advance as an email attachment. I can't do the same on a disk because the disks I have are read-only, and if any changes or updates were to occur afterward, I would need to save the whole file(s) all over again which would take up more room.

Alas, there have been so many great threads on that board that I should have saved somewhere but wasn't able to. I have only my memory of them now. As for this blog post, I should finish it up here before it gets too long.

Thanks for reading,

Cerne 08:32, September 7, 2010 (UTC)