Co-orbiting Earths

As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be unstable
for Earth-sized planets orbitting the Sun at about 1 AU? If that
would be unstable (perhaps due to some tidal effect or something)
what would be the largest planets that would be stable?

If they are stable, what distance apart would the two orbits be?
And how long between each swapping?

Also, how close in mass do the two have to be? If things had
worked out differently, could the Mars-sized planet that helped
make the Moon have gotten into this kind of arrangement with
Earth or would it have been too small?

--
Dan Tilque

Dan Tilque wrote:

As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be unstable
for Earth-sized planets orbitting the Sun at about 1 AU? If that
would be unstable (perhaps due to some tidal effect or something)
what would be the largest planets that would be stable?

The earth and moon, e.g. are certainly in a co-orbit. The thing
is that they are TOO stable. That is, they are very tightly bound
and the tidal forces of the sun's central force field are
a perturbation on the earth-moon orbit.

If they are stable, what distance apart would the two orbits be?
And how long between each swapping?

The tidal, that is differential, force in a free fall zone
around a central body is given by GM/r^3 * a, where a is the
separation from the free fall point. But note GM/r^3 gives
omega^2, by Kepler's third law, so the scale of the tidal force
is very easily given by the orbital period. In our case we
have to compare 1 year ( or 2pi/1 year, squared ) with 0.69 days,
for Janus and Epimetheus.

What we want is a gravitational attraction in the reduced mass
system of the co-orbiting bodies to be comparable to the tidal
force at the same distance:

omega_1^2 * a_1 / ( G mu_1/a_1^2) = omega_2^2 * a_2 / ( G mu_2/a_2^2)

(a_2/a_1)^3 = (omega_1/omega_2)^2 * (mu_2/mu_1)

"2" is earth/moon and "1" is Janus/Epimetheus

so

a_2 = 50 km * ( (365/.7)^2 * (1/7.0e-8) )^ 1/3 =

50 km * 15720 = 786000 km

so according to my calculations, if the moon were about 3 times
as far away as it is now, the earth and moon should be capable
of showing "co-orbital" behavior.

Note that the Janus/Epithemeus attraction 50km is still
substantially larger than the the tidal forces of Saturn's
gravitational field, and of course that ratio is duplicated
for the earth/moon system by my formula.

I think the "swapping time" is very touchy, and that it's
the result of what amounts to a very eccentric orbit in
the heavily perturbed field. I think it's somewhat analogous
to a comet return time.

Note that the mass ratio of Janus/Epithemeus is 2.01/0.56 = 3.6,
so if you believe in Center of Mass, I think you have to accept
this "swapping" as a heuristic.

Also, how close in mass do the two have to be? If things had
worked out differently, could the Mars-sized planet that helped
make the Moon have gotten into this kind of arrangement with
Earth or would it have been too small?

I think to a reasonable approximation you can treat any
co-orbiting pair as a reduced mass problem in the presence
of the tidal field.

Please take many and abject disclaimers as understood.

Lew Mammel, Jr.

Lewis Mammel wrote:

Please take many and abject disclaimers as understood.

Not abject enough! I got the whole setup wrong. I'm thinking
wait a minute, 50 km ?? DOH! . Anyway, the basic approach still
stands. If you take the tidal force and add the centrifugal
force due to a rotating frame, you get ( -2, +1 ) + ( -1, -1) = ( -3, 0 )
meaning there is no force along the orbit. Test particles lined
up along the orbit are stationary in the rotating frame as the
follow the circular orbit.

The first order term as you move in or out is sufficient to treat
objects which do not stray far from the circular orbit. You
just have to apply a Coriolis term. An object in a slightly
smaller circular orbit has the Coriolis term exactly cancel
the tidal term. If you add a small acceleration term along its
orbit, due to attraction of another objec in the orbit, it will
gain speed and veer outward as the coriolis term increases.

I tried some calculations with a simple linear acceleration
along the orbit, and found that for small values of the acceleration
I actually reproduced the low-to-high orbit transfer.

I think there's a whole range of values where this can happen.
I think the condition is that the delta v for the transfer has
to be small enough that it happens without overtaking the
other object too closely.

BTW, the encounter cycle time is just the reciprocal delta omega
time, from Kepler's 3rd Law.

d(w^2 r^3) = 0

2w r^3 dw + w^2 3r^2 dr = 0

dw/w = -3/2 dr/r

dw = 2pi/( .69 days ) -3/2 50km / 150000km = 2pi/( 3.8 years )

"Dan Tilque" wrote in message
...
As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be unstable
for Earth-sized planets orbitting the Sun at about 1 AU? ...

Also, how close in mass do the two have to be? ...

That's the easy part, the answer is "not very". Consider
Earth and Cruithne. The details of the orbits depend on
the ratio of the masses of course.

"Lewis Mammel" wrote in message
...
....
I think there's a whole range of values where this can happen.
I think the condition is that the delta v for the transfer has
to be small enough that it happens without overtaking the
other object too closely.

I think the question is effectively how massive could
Cruithne be. Given that Epimetheus and Janus are
similar in mass, I don't see why the situation would
be unstable if Cruithne were similar in mass to Earth,
though I guess they both need to be small in comparison
to the Sun.

All of this is guesswork of course Lewis, your quantitative
approach is much more informative.

George

George Dishman wrote:

"Dan Tilque" wrote in message
...
As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be
unstable for Earth-sized planets orbitting the Sun at about 1
AU? ...

Also, how close in mass do the two have to be? ...

That's the easy part, the answer is "not very". Consider
Earth and Cruithne. The details of the orbits depend on
the ratio of the masses of course.

Good point, but I was thinking of two planets in more circular
orbits. Cruithne is in a rather eliptical orbit.

Actually what I'm really wondering is if you could have two
inhabited (or inhabitable) planets in such orbits. Highly
eliptical orbits do not work for that purpose. I also wonder if
anyone has used this idea in an sf story (crossposting to
rec.arts.sf.science for this question).

Googling Cruithne, I found an interesting page (link below). If
you look in the Stable Figure-Eight Orbits page, you'll see a
simulation of three planets in intertwined orbits about a sun.
Very interesting, although unlikely to occur naturally.

http://burtleburtle.net/bob/physics/index.html

--
Dan Tilque

George Dishman wrote:

All of this is guesswork of course Lewis, your quantitative
approach is much more informative.

Thank you. I've done a little more and get some interesting
results. I can do a finite difference simulation in the
rotating frame of a circular orbit with these equations:

x1 = 30+x+vx*DT/2 ;
y1 = y+vy*DT/2;
r = sqrt( x1*x1 + y1*y1 );

x += vx*DT;
y += vy*DT;
vx += ( 2.0*vy - A*x1/r/r/r) * DT;
vy += ( -2.0*vx + 3.0*y1 -A*y1/r/r/r) *DT;

This puts the co-orbital attractive center at x = -30,
I used initial conditions of x0 = vy0 = 0, y0 = -1, vx0 = -1.5

The interesting thing is that the distance scale is arbitrary,
as long as it is small compared to the orbital radius. It doesn't
matter if y = - 1 mile or -100 miles. The unit of time is T/2pi = 1/omega,
the orbital period divided by 2pi. The coriolis terms is 2 omega X v, and
the centrifugal+tidal term is 3 omega^2 y. With A=0 and the
stated initial conditions, vx and vy are constant, representing
a co-orbiting body in a lower orbit.

If you leave A=0 and vary the initial conditions, you get periodic
results for nearly circular orbits. Nonzero A gives various relative
strengths of the coorbital interaction. Note A = G mu/omega^2, or
(omega'/omega)^2, where omega' is 2pi/( orbital period of coorbital
objects in circular oribt at unit distnce ). Here's a table of results:

A t y vx vy ymin xmin

1.000 37.638 1.71577 2.82933 -0.27618 0.0843 -28.1426
2.000 42.200 1.03664 1.57046 0.08664 0.0122 -25.7702
3.000 41.760 1.00862 1.51688 0.03161 0.0046 -23.8838
4.000 40.802 0.99592 1.49166 0.02268 0.0035 -22.2817
5.000 39.639 0.98842 1.47674 0.01259 0.0029 -20.8866
6.000 38.395 0.99552 1.49103 0.00097 0.0008 -19.6602
7.000 37.158 1.01262 1.52517 0.00521 -0.0026 -18.5809
8.000 35.960 1.01747 1.53462 0.02541 -0.0053 -17.6304
9.000 34.790 1.00128 1.50204 0.03925 -0.0059 -16.7851
10.000 33.635 0.98206 1.46372 0.02947 -0.0046 -16.0223
11.000 32.514 0.98469 1.46928 0.00746 -0.0021 -15.3244
12.000 31.466 1.00834 1.51665 0.00060 0.0010 -14.6790
13.000 30.501 1.02986 1.55933 0.01787 0.0043 -14.0776
14.000 29.600 1.03157 1.56213 0.04593 0.0073 -13.5141
15.000 28.736 1.01223 1.52299 0.06543 0.0099 -12.9847
16.000 27.889 0.98457 1.46770 0.06342 0.0114 -12.4871
17.000 27.057 0.96759 1.43424 0.04108 0.0113 -12.0208
18.000 26.261 0.97334 1.44638 0.01415 0.0083 -11.5878
19.000 25.519 0.99777 1.49553 0.00026 0.0007 -11.1936
20.000 24.840 1.02695 1.55363 0.00614 -0.0084 -10.8437


x is always, zero, which means that the object has fallen as far towards
the co-orbiter center of attraction as it can ( xmin ) and returned to
the relative starting point. Note that the interactions take several
revolutions of the orbit and are relatively stable for A = 2.0 .

I take ymin as a figure of merit for the stability of the interaction,
since if y=0 at x=minimum, there is dynamical symmetry between the
approaching and receding leg of the interaction. If you end up with
y_final= 1.0, vx_final= 1.5, then you have a perfectly circular orbit
higher and slower than the approaching lower and faster orbit.

But just now I calcuate A= 120 for J/E ! I had thought it was
smaller for some reason. I start to see more instability for
interaction that large, but I can hardly take my model as definitive.
Anyway, I hope my qualitative result, that the phenomenon occurs
over a broad range of scales, is accurate.

So, I claim the phenomenon can qualitatively take place for earthlike
objects in earthlike orbit, but I think stability is an issue. This is
still a theoretical question for Janus/Epimetheus, I think, and it may be
that the oblateness of Saturn contributes to their stability -
but I don't know!

Lew Mammel, Jr.

"Dan Tilque" wrote in message ...
George Dishman wrote:

"Dan Tilque" wrote in message
...
As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be
unstable for Earth-sized planets orbitting the Sun at about 1
AU? ...

Also, how close in mass do the two have to be? ...

That's the easy part, the answer is "not very". Consider
Earth and Cruithne. The details of the orbits depend on
the ratio of the masses of course.

Good point, but I was thinking of two planets in more circular
orbits. Cruithne is in a rather eliptical orbit.

Actually what I'm really wondering is if you could have two
inhabited (or inhabitable) planets in such orbits. Highly
eliptical orbits do not work for that purpose. I also wonder if
anyone has used this idea in an sf story (crossposting to
rec.arts.sf.science for this question).

Googling Cruithne, I found an interesting page (link below). If
you look in the Stable Figure-Eight Orbits page, you'll see a
simulation of three planets in intertwined orbits about a sun.
Very interesting, although unlikely to occur naturally.

http://burtleburtle.net/bob/physics/index.html

The more relevant link there for this thread is

http://burtleburtle.net/bob/physics/kempler.html

which mentions Saturn's co-orbiting moons (using a simulations whose
numbers I just made up, but demonstrates the effect). It goes on to
simulate 2, 6, 24, and 48 co-orbiting earths of slightly off masses
positions and velocities plus with Jupiter orbiting in its usual
place. 48 earths falls apart but the others hold together. It
finishes off with a simulation of Niven's puppeteer system (a
Klemperer rosette of five planets with no central sun).

As for masses, these systems do springy rebounds. Just like in
gasses, the smaller an object's mass, the more energy it has. The
more chaotic the system, the more equal the masses have to be,
otherwise the little masses pick up too much energy and get thrown out
of the system.

Bob Jenkins wrote:

The more relevant link there for this thread is

http://burtleburtle.net/bob/physics/kempler.html

which mentions Saturn's co-orbiting moons (using a simulations whose
numbers I just made up, but demonstrates the effect). It goes on to
simulate 2, 6, 24, and 48 co-orbiting earths of slightly off masses
positions and velocities plus with Jupiter orbiting in its usual
place. 48 earths falls apart but the others hold together. It
finishes off with a simulation of Niven's puppeteer system (a
Klemperer rosette of five planets with no central sun).

The simulations make it very clear that the 24- and 5-body cases are
unstable. Also, if you zoom out from the 48-body example, it's clear
there's an additional distant body involved, and it's unclear whether or
not that's an intended part of the simulation or what effect it has on
general stability.

--
Erik Max Francis && && http://www.alcyone.com/max/
San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis
We're here to preserve democracy, not to practice it.
-- Capt. Frank Ramsey

"Dan Tilque" wrote in message
...
George Dishman wrote:

"Dan Tilque" wrote in message
...
As most of you know, Saturn's moons Epimetheus and Janus
co-orbit. Their orbits are about 50 Km apart which they swap
about every 4 years.

Is there any reason that this kind of co-orbit would be
unstable for Earth-sized planets orbitting the Sun at about 1
AU? ...

Also, how close in mass do the two have to be? ...

That's the easy part, the answer is "not very". Consider
Earth and Cruithne. The details of the orbits depend on
the ratio of the masses of course.

Good point, but I was thinking of two planets in more circular
orbits. Cruithne is in a rather eliptical orbit.

OK but it still shows they don't have to be close
in mass. If you are looking for a circular orbit,
there is also "2002 AA29" (whose name I couldn't
remember) which is in a more circular orbit than
the Earth:

http://www.astro.uwo.ca/~wiegert/AA29/AA29.html

The complex diagram is because it is drawn in a
co-rotating frame but each small loop also takes
a year. Their Cruithne page is he

http://www.astro.uwo.ca/~wiegert/3753/3753.html

Actually what I'm really wondering is if you could have two
inhabited (or inhabitable) planets in such orbits. Highly
eliptical orbits do not work for that purpose.

It seems entirely possible, and with the orbiting
triplet you found below, they could have some fun
predicting eclipses!

I also wonder if
anyone has used this idea in an sf story (crossposting to
rec.arts.sf.science for this question).

Googling Cruithne, I found an interesting page (link below). If
you look in the Stable Figure-Eight Orbits page, you'll see a
simulation of three planets in intertwined orbits about a sun.
Very interesting, although unlikely to occur naturally.

http://burtleburtle.net/bob/physics/index.html

Fascinating, I didn't know that was possible. Thanks
for the pointer.

George

Erik Max Francis wrote in message ...
Bob Jenkins wrote:

http://burtleburtle.net/bob/physics/kempler.html

The simulations make it very clear that the 24- and 5-body cases are
unstable. Also, if you zoom out from the 48-body example, it's clear
there's an additional distant body involved, and it's unclear whether or
not that's an intended part of the simulation or what effect it has on
general stability.

My goal in including Jupiter (and making the positions, masses,
velocities of the Earths slightly off) was to make it less stable.
Everything I tried without a central sun was very unstable anyhow so I
didn't include a Jupiter. The 24-earth simulation with a central sun
seems stable despite Jupiter and the errors. Although perhaps I just
didn't run it long enough. The 48-earth isn't stable (with the same
Jupiter and errors). Reducing the earth masses by 2/3 made the
48-earth simulation stable (again, stable for longer, maybe unstable
if I'd run it longer still).

The simulator used seems adequate for the task. I've tried 11th
order, 9th order, 7th order integration methods (all symmetric
multistep methods though). These simulations behave roughly the same
with all of them. So it seems unlikely that these behaviors are an
artifact of the simulation method.