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Special Ratio problem (not a math problem. It's the physic problem)!

  1. Nov 17, 2012 #1
    1. The problem statement, all variables and given/known data

    The binary star system Krell consists of two stars, Krell A and Krell B, orbiting each other, with no other ojects nearby. Assume Krell A, the smaller star of mass mA, moves in a circle of radius rA at speed vA. Krell B, the larger star of mass mB = 2.51mA), moves in a circle of radius rB at speed vB. The two stars are separated by a distance d.

    Find the ratio of the periods of orbit TA/TB

    2. Relevant equations

    I would assume these formulas:

    →T²/r³
    →T²/r³ = 4π²/(G(M + m))

    3. The attempt at a solution

    I believe the approach is...

    T_A²/R_A³ = T_B²/R_B³
    T_A²/T_B² = R_A³/R_B³
    (T_A/T_B) = √((R_A/R_B)³)

    From the previous part of the problem, I've found that R_A/R_B = 2.51. Then, I substitute that for the above expression and got the answer incorrect.

    T_A/T_B = √(2.51³) ≈ 3.98

    But the answer is incorrect.
     
  2. jcsd
  3. Nov 17, 2012 #2

    Borek

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    In a few days from now Sun will enter the Sagittarius. How long till it will enter the Sagittarius again?

    From the Sun point of view, Earth will - again in a few days - enter the Gemini. How long till the Earth will be visible in the same direction (on the background of the distant stars) from the Sun?
     
  4. Nov 17, 2012 #3
    Ah... Seems like there is no way to figure that out. :D

    Now, help me with the question, or I will wait for someone to assist me.
     
  5. Nov 17, 2012 #4

    gneill

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    Maybe approach the problem from a different perspective... It's an isolated system of two objects that don't interact with anything external. What characteristics can you ascribe to the center of mass of such a system?
     
  6. Nov 17, 2012 #5
    The masses are distance d apart, orbiting the common center of the mass. This is by the modified form of Kepler's third law.
     

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  7. Nov 17, 2012 #6

    gneill

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    Where does the center of mass lie with respect to the orbiting bodies (in terms of the geometry of the system)? Does it move?
     
  8. Nov 17, 2012 #7
    The center of mass lies along the path where the larger mass will rotate. I believe it moves.
     
  9. Nov 17, 2012 #8

    gneill

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    Umm, no. If you have two masses and you want to find the center of mass of the pair, do you expect to find it to the left of them both? Above them both? Or perhaps along the line segment joining the two?

    If no external forces are acting on the system, what would cause the center of mass to move? This is introductory stuff regarding the motion of the center of mass of an isolated system (relies on conservation of momentum).
     
  10. Nov 17, 2012 #9
    Then, the center of mass lies along the line segment joining the two? I believe that the center lies somewhere closer to the larger mass.
     
  11. Nov 17, 2012 #10
    Nvm. The answer is 1. The velocity and the radius of the orbits are inversely proportional to each other. If the distance between the masses changes, then the periods for both masses are not all the same. If both masses are distance d apart, orbiting the common center of mass at the opposite directions, then the center of mass is "consistent" and so the change of momentum is 0.
     
  12. Nov 17, 2012 #11

    gneill

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    Correct conclusion, but part of your reasoning is flawed. Even if the distance between the bodies changes, so that the orbits are elliptical rather than circular, the periods must still be the same.

    The center of mass cannot move (assuming the coordinate system chosen has its origin at the center of mass) because there are no external forces operating. Further, the center of mass must always lie along the line joining the two mutually orbiting bodies. That means that when one body completes a full orbit and returns to its starting position, the other body must also return to its diametrically opposed position at the same instant.
     
  13. Nov 18, 2012 #12

    Borek

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    Now, if you will try to understand the geometry I described in my post, you will see it was leading to this conclusion.

    It is just a matter of understanding what the orbital period is - if you are on one celestial body and observing other, orbital period is a time required for the other body to appear in exactly the same place on the sky (in the reference frame of distant stars, which makes local motion of the system irrelevant). That means after a period passes from the body which you are observing you are also seen in exactly the same place (which happens to be in exactly the opposite direction, hence zodiac sings that are exactly 6 months apart).
     
  14. Nov 18, 2012 #13
    Good advice!

    How much do you know about astrology and stuff like this? How do you know such thing?
     
  15. Nov 18, 2012 #14

    Borek

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    Nothing. All I know is that there are 12 zodiac signs and they are placed more or less on the ecliptic.
     
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