Calculating orbital velocities of binaries?

Hi - -

I need real help on what I hope is an elementary problem in stellar
binary motion. For simplicity's sake I imagine a pair of stars in a
stable, perfectly circular orbit. Their masses are M1 and M2. Their
center-of-mass distance is R and their respective distances to that
center-of-mass are r1 and r2, ie: R = r1 + r2. I want to know two
things:

(A) What relations correctly calculate the stars' respective
velocities
(I suppose one would say in the center-of-mass frame?)

(B) What, if any, corrections from Special Relativity might one need
to
insert for orbital velocities beginning to approach the relativistic
regime?
Also, for the sake of easy discussion, I omit any issues raised by
General Relativity.

Regarding (A), I had attempted a derivation following the clues given
in an old text, "The Physical Universe: An Introduction to Astronomy",
by Frank Shu. My result for the orbital velocity v1 for M1 was

(1) (v1)^2 = (r2 / R) x G(M1 + M2)/R ,

where G is the Newtonian gravitational constant.

However, when I later used the information given on the Hyperphysics
site on binary motion (hosted by Georgia State Univ.) at

http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#bo

their relation "From the gravity force and the necessary centripetal
force":

(2) GM1M2 / R^2 = M1(v1)^2 / r1 ,

yields a different result when solving for v1 :

(3) (v1)^2 = r2 / R x GM2 / R .

Note that M1 does not enter into the right hand side. I would really
appreciate any help. What am I doing wrong? :-\

Thanks very much,
Gene

stargene wrote:
I need real help on what I hope is an elementary problem in stellar
binary motion. For simplicity's sake I imagine a pair of stars in a
stable, perfectly circular orbit. Their masses are M1 and M2. Their
center-of-mass distance is R and their respective distances to that
center-of-mass are r1 and r2, ie: R = r1 + r2. I want to know two
things:

(A) What relations correctly calculate the stars' respective
velocities
(I suppose one would say in the center-of-mass frame?)

Following the principles of Socratic dialog, I'm not going to answer
your question, but instead suggest a route you can follow to work the
answer out for yourself: (My copy of Shu's book is buried right now,
but I suspect this is very similar to the derivation he suggests.)
(1) If you know the masses M1 and M2, what can you say about the
relationship between the radia-from-the-center-of-mass r1 and r2?
(2) To simplify the algebra later, try to
(a) write r1 as a function of r1+r2 and the masses, and
(b) write r2 as a function of r1+r2 and the masses.
(3) What is M1's acceleration in its orbit? To simplify the algebra,
try to use your results from (2) to express this in terms of the
orbital frequency (measured in radians/second) and r1+r2.
(4) Now look at the dynamics: What's the gravitational force acting
on M1?
(5) What does Newton's 2nd law then say about M1's motion?
(6) Can you solve the equations you get in (5) for the orbital frequency
in terms of the Newtonian gravitational constant G and the masses?
(7) As a check, you could repeat (3) through (6) using M2 instead of M1
and verify that you get the same orbital frequency.
(8) Can you express the orbital velocities in terms of the orbital
frequency and other things you already know?


[[...]]
Regarding (A), I had attempted a derivation following the clues given
in an old text, "The Physical Universe: An Introduction to Astronomy",
by Frank Shu.

This was and remains a superb text. I think it may have a 2nd edition
now (probably with a co-author).

My result for the orbital velocity v1 for M1 was

(1) (v1)^2 = (r2 / R) x G(M1 + M2)/R ,

[[...]]
However, [[using some equations from a nice physics web site]]
yields a different result when solving for v1 :

(3) (v1)^2 = r2 / R x GM2 / R .

Hint: What if one of the masses is much smaller than the other one?
For example, what if M2 = the mass of our Sun, while M1 = the mass
of one of the planets in our solar system? (Jupiter, the most massive
planet, is about 0.1% of the mass of the Sun.) What approximations
can you reasonably make?

--
-- "Jonathan Thornburg [remove -animal to reply]"
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

stargene ) writes:
Hi - -

I need real help on what I hope is an elementary problem in stellar
binary motion. For simplicity's sake I imagine a pair of stars in a
stable, perfectly circular orbit. Their masses are M1 and M2. Their
center-of-mass distance is R and their respective distances to that
center-of-mass are r1 and r2, ie: R = r1 + r2. I want to know two
things:

(A) What relations correctly calculate the stars' respective
velocities
(I suppose one would say in the center-of-mass frame?)

(B) What, if any, corrections from Special Relativity might one need
to
insert for orbital velocities beginning to approach the relativistic
regime?
Also, for the sake of easy discussion, I omit any issues raised by
General Relativity.

Regarding (A), I had attempted a derivation following the clues given
in an old text, "The Physical Universe: An Introduction to Astronomy",
by Frank Shu. My result for the orbital velocity v1 for M1 was

(1) (v1)^2 = (r2 / R) x G(M1 + M2)/R ,

where G is the Newtonian gravitational constant.

However, when I later used the information given on the Hyperphysics
site on binary motion (hosted by Georgia State Univ.) at

http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#bo

their relation "From the gravity force and the necessary centripetal
force":

(2) GM1M2 / R^2 = M1(v1)^2 / r1 ,

yields a different result when solving for v1 :

(3) (v1)^2 = r2 / R x GM2 / R .

Note that M1 does not enter into the right hand side. I would really
appreciate any help. What am I doing wrong? :-\

I'm not sure how you got eq (1), but I'll sketch how I'd derive (2) and
(3). The gravitational force between the two stars is, according to Newton:

F = G*M1*M2/R^2.

Also according to Newton, this is the force acting on either star. Since
we have circular orbits, we can equate this force to the centripetal force
required to hold either star in circular motion. For star 1

F= M1*v1^2/r1 (X)

(Similarly for star 2: F = M2*v2^2/r2 .)

Substituting for F in eq (X) gives eq (2) and rearrangement gives eq (3a):

v1^2 = G*M2*r1 / R^2 (3a)

(** NB: r1 not r2)

Note that is is really an expression of acceleration: v1^2 / r1 is the
centripetal acceleration of star 1 in its circular orbit--this acceleration
depends on the mass M2 pulling on star 1 and on how far star 1 is from the
centre of its circle. M1 doesn't appear directly (it has a hidden influence in
that it determines the size of r1) for essentially the same reason that heavy
objects and light objects on earth fall at the same rate (in vacuum).

--John Park

stargene ) writes:
Hi - -

I need real help on what I hope is an elementary problem in stellar
binary motion. For simplicity's sake I imagine a pair of stars in a
stable, perfectly circular orbit. Their masses are M1 and M2. Their
center-of-mass distance is R and their respective distances to that
center-of-mass are r1 and r2, ie: R = r1 + r2. I want to know two
things:

(A) What relations correctly calculate the stars' respective
velocities
(I suppose one would say in the center-of-mass frame?)

(B) What, if any, corrections from Special Relativity might one need
to
insert for orbital velocities beginning to approach the relativistic
regime?
Also, for the sake of easy discussion, I omit any issues raised by
General Relativity.

Unless you are working to extremly high accuracy, I don't think you need
worry about Special Relativity for normal stars in normal orbits. You'd
need obital periods of a few minutes to show any real effects--which would
correspond to neutron stars a few thousand km apart. For such a system I
don't know that you could ignore General Relativity.


Regarding (A), I had attempted a derivation following the clues given
in an old text, "The Physical Universe: An Introduction to Astronomy",
by Frank Shu. My result for the orbital velocity v1 for M1 was

(1) (v1)^2 = (r2 / R) x G(M1 + M2)/R ,

where G is the Newtonian gravitational constant.

However, when I later used the information given on the Hyperphysics
site on binary motion (hosted by Georgia State Univ.) at

http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#bo

their relation "From the gravity force and the necessary centripetal
force":

(2) GM1M2 / R^2 = M1(v1)^2 / r1 ,

yields a different result when solving for v1 :

(3) (v1)^2 = r2 / R x GM2 / R .

Note that M1 does not enter into the right hand side. I would really
appreciate any help. What am I doing wrong? :-\

It might be helpful to express r1 and r2 in terms of R and the two masses
(and see Jonathan Thornburg's post).

--John Park

Socratic dialog... I was afraid of this. You should probably just e-
mail
me the hemlock right now. I'm retired, an amateur and within inches
of seventy, and my normal bumfuzzlement has increased due to the
hard economic times for my wife and myself. Besides, Shu's spare
hints are what got me in trouble in the first place. Nevertheless
I'll
give it a whirl.

(Frank Shu also does not believe in presenting solved problems in his
book, a touching act of faith (or sadism) which nearly drove me
around
the bend at times.)

Below I give my derivation of my eq (1) using (or misusing) the
'Socratic'
hints given by Shu in his original chapter on binary stars, since he
never actually directly discusses the orbital velocities, let alone
cal-
culates them (pgs. 182-3 of the 1982 edition):

He presents, for a binary pair M1, M2 in circular orbit where M1M2,

(a) r1 = [M2/(M1 + M2)] R
(b) r2 = [M1/(M1 + M2)] R

(again, R = r1 + r2),

without proof. But (a) follows by knowing, as you hinted, that

(c) M1r1 = M2r2 .

Intuitively, this is analogous to Archimedes' 'law of levers'. (a)
follows
by adding M2r1 to both sides of (b) and then factoring out r1 and M2,
then solving for r1. Similarly for (b).

Shu then states that

(d) v1 = Omega x r1 ,

where Omega (= 2pi / period of revolution) is the angular speed of
revolution. And analogously, v2 = Omega x r2 .

He then asks us to prove that

(d) (Omega)^2 = G(M1 + M2) / R^3 .

My proof (which parallels your hint about Newton's 2nd law) is :

Newton's def. of force F is F = Ma, 'a' being acceleration. The
accel-
eration a1, felt by M1, is (v1)^2 / r1. a2 for M2 is (v2)^2 / r2 .
Thus

(e) F1 = M1 (Omega x r1)^2 / r1 .

Substituting for r1 the right side of (a) gives

(f) F1 = M1 x (Omega)^2 x M2R / (M1 + M2) .

We know that F1 = M1a1 = GM1M2 / R^2 , (this satisfies your hint
# (4)), and combining this with (e) gives

(g) M1(Omega)^2 x M2 R / (M1 + M2) = GM1M2 / R^2 . So Shu's
statement (d) about Omega is true.

The individual factors M1 and M2 cancel out. Solving for (Omega)^2
gives Shu's rel. (d).

Now, taking (d) as the definition for velocity v1 for mass M1, and
squaring both sides, gives

(h) (Omega)^2 = (v1 / r2)^2 .

I now can equate the right side of (h) with the right side of (d).
This
gives

(i) (v1 / r1)^2 = G(M1 + M2) / R^3 .

Solving for (v1)^2 gives

(j) (v1)^2 = (r1)^2 x G(M1 + M2) / R^3 .

Taking the square root of both sides gives

(k) v1 = (r1 / R) x sqrt [G(M1 + M2) / R ] .

Or alternatively, combining (a) and (k), I can recast v1 as

(l) v1 = M2 / (M1 + M2) x sqrt [G(M1 + M2) / R ] .

Now if M1 M2, eg: M1 is a NASA satellite and M2 the mass
of the Earth, then (l) can reduce for most purposes to

(m) v1 = sqrt [G(M2) / R ] .

Is this what you were hinting with your final question?

Finally, I still don't know how to decide between my (k) and
the different relation I deduced from the Hyperphysics site
where

(3) (v1)^2 = r2 / R x GM2 / R , or: (r2 / R^2) x GM2

from my original post. I only know that with M1M2,
both equations reduce to (m) as approximations. I can
only guess that either my (l) is incorrect due to some false
assumption, or that both equations may be correct but
in different frames of reference. That stretches my brain
to its limit. "A poor one, but mine own."

I hope that Socrates will take pity on me and provide some
meager explanation. Also my second question still stands,
since I cannot decide what, if any, Special Relativity correction
might be true in principle, even when orbital velocities are
well below the relativistic regime and tidal processes can
be ignored.

Thanks,
Gene

I am just posting a typo correction here to (3). It should read

(3) (v1)^2 = r1 / R x GM2 / R ,

and not:

However, [[using some equations from a nice physics web site]]
yields a different result when solving for v1 :

(3) (v1)^2 = r2 / R x GM2 / R .

for what it's worth,
stargene

Actually, yes, for my own arcane reasons, testing a half-vast model I
tinker
with, I am working to extremely high accuracy. Also, thank you for
the
correction. I had made a math typo error in my haste. My (3) now
should
read

(v1)^2 = (r1 / R) x (GM2 / R)

Gene

Unless you are working to extremly high accuracy, I don't think you need
worry about Special Relativity for normal stars in normal orbits. You'd
need obital periods of a few minutes to show any real effects--which would
correspond to neutron stars a few thousand km apart. For such a system I
don't know that you could ignore General Relativity.

stargene ) writes:
Actually, yes, for my own arcane reasons, testing a half-vast model I
tinker
with, I am working to extremely high accuracy. Also, thank you for
the
correction. I had made a math typo error in my haste. My (3) now
should
read

(v1)^2 = (r1 / R) x (GM2 / R)

Gene

Unless I'm much mistaken, if you now substitute for r1 in terms of the two
mmasses and R, you should see your problem disappear.

--John Park