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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 |
#2
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Calculating orbital velocities of binaries?
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 |
#3
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Calculating orbital velocities of binaries?
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 |
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Calculating orbital velocities of binaries?
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 |
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Calculating orbital velocities of binaries?
Socratic dialog... I was afraid of this. You should probably just e-
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 |
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Calculating orbital velocities of binaries?
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 |
#7
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Calculating orbital velocities of binaries?
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. |
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Calculating orbital velocities of binaries?
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 |
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