Expert: Jayendra Upadhye - 10/25/2004

Question
There are a certain solar system which contains;
1. two similar stars with approximately equal level of gravity
2. some planets
3. the stars moves around the joint centre of gravity which lies everywhere in the space on the section between them (but not inside one of them)
My questions are:
1. Is existing of planets possible in this system?
2. (If yes)What is a shape of planet's orbit in this system?
3. Is existing of eco-sphere possible in this system?  

Answer
Hi,
1 - If both stars have approx equal mass, and are separated by distance d and have mass say m, then the center of gravity will be (assuming the attendent planets have "very insignificant" comparable mass) almost d/2 units from either stellar center.
IF YOU ARE IN A HURRY GO DIRECTLY TO "YOUR ANSWER" AT BOTTOM.ELSE BREOWSE ON!
Placing the center og gravity / center of mass of the system roughly midway between the two stars.

2 - To an external observer stationed "above" the orbital plane, and along the orbital axis passing thru that center, both stars will appear to move on opposing sides of a circle of diameter d units.

3 - To an observer at either stellar center, however, the "other" star will appear to be orbiting ones "own" star with an orbital radius d units ie orbital circumference 2d units!
[though this appears strange at first, think about it calmly and you will agree!]

4 - Any planet at distance R units, from the stellar center of gravity and with instantaneous linear velocity V units, [at right angles to the line connecting it to the C.G.] will describe a perfect circular orbit when and if..
{(V * V)/r} = G*(2*M)/(r*r) -> eqn 1
Where G is universal constant of gravitation.
and 2*M is total gravitating mass of the 2 stars.

This is so because force due to gravity on a planet of mass m units is f = G*(2*M)*m/(r*r) -> eqn 2
where f = force and m is mass of planet adn 2*m is gravitating central mass.

but f/m = g is gravitational acceleration on the planet at its distance r units.
but this g is ALSO the centripetal acceleration in the uniform circular motion of the planet about the center of gravity (C.G)
We know in uniform circular motion
centripetal acceleration = m * (V*V)/r -> eqn 3 or

combining eqn 3 and 2 and eliminating m and r
V squared = 2*G*M/r or
V = sqrt{2*G*M/r}     -> eqn 4

Simply put, The orbital velocity of a planet in uniform circular orbit will fall as the inverse of the root of its orbital radius.

[This is also in keeping with observation as inner planets have to move faster to counter the greater force of gravity by "centrifugal" tendencies].

Most importantly, stellar masses in close binaries and triple systems, behave as though they were a virtual star placed at the stellar C.G., and with mass = combined mass.

The planets would have to orbit at distances greater than swath likely to be cut by either star's surface as it moves along its "own" orbit.
Else ofcourse the planet would end up grazing the star's atmosphere!

I have put a couple beautiful orbit simulators at the qb4all.com qucik basic site. these programs use the qbasic 4.5 code and read .orb files to simulate orbital motions of upto 10 stars of identical mass, randomly placed and of random initial velocities.
it is a download site. for orbit7.zip (executable plus orbit files, the link is http://qb4all.com/files/download.asp?fileid=847
for orbit3 binary star simulator, the link is http://qb4all.com/files/download.asp?fileid=606
copy these lines into a file called circle1.orb and run orbit3.exe on this file (by typing y and furnishing circle1.orb as the name of file to import).
see how the program plots the binary system on your pc screen. it even furnishes orbital eccentricity when one full orbit is completed.
contents of circle1.orb file:-
2
380
240
14
0
.129099
0
0
260
240
14
0
-.129099
0
0
happy surfing
************ YOUR ANSER **********
1. Is existing of planets possible in this system?
CERTAINLY!
2. (If yes)What is a shape of planet's orbit in this system?
like in any other planetary system, it would be an ellipse or a circle (a particular case of an ellipse!). but with closest approach to C.G wider than the stellar orbital dia + half the star's own diameter! to avoid collision!
3. Is existing of eco-sphere possible in this system?  
certainly as long as the orbit is not overtly cigar shaped and is rather round, and the planet is in the "habitable zone" where the surface temperature and green house effect is balanced at our bio-sphere levels.
Only the guys there would daily observe the two suns "rise" simultaneously or a bit one after the other and setting in that order! there would be ofcourse mythologies about "solar twins", "twin gods in love with each other" and so on! And astrologers would fight pitched battles over which "sun" was more benefecient to it's subjects!
jayen