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The Time to Fall in a Plunge Orbit

  1. May 28, 2016 #1
    1. The problem statement, all variables and given/known data
    Two masses, M and m, are initially (at time t=0) at rest with respect to each other and separated by a distance d. They are affected only by their mutual gravitational attraction. Find the amount of time elapsed beginning at the instant the separation is equal to d/2 and ending at the instant the separation is equal to d/3. (Ignore relativity.)

    2. Relevant equations
    The Vis Viva equation is relevant as a starting point. Solve it for the speed in orbit, and, since the orbit is a plunge orbit, change v into the differential dr/dt, creating an ordinary non-linear differential equation. This ODE is to be solved to give time elapsed from t=0 as a function of r, such that 0 ≤ r ≤ d.

    3. The attempt at a solution
    This was a homework problem that I was given as an undergraduate, along with those who were my classmates at the time. We solved it, but it was somewhat more challenging that we'd at first expected. The solution is

    t(r) = √{d/[2G(M+m)]} {√(rd−r²) + d Arctan[√(d/r−1)]}

    t(d/2) = (1/2 + π/4) √{d³/[2G(M+m)]}

    t(d/3) = {√(2)/3 + Arctan[√(2)]} √{d³/[2G(M+m)]}

    Δt = {√(2)/3 + Arctan[√(2)] − 1/2 − π/4} √{d³/[2G(M+m)]}

    Δt ≈ 0.141322975518 √{d³/[2G(M+m)]}

    From t(r), you may notice that when r«d then, effectively, r ≈ 0, and

    t ≈ π √[d³/(8GM)]

    This approximate solution can be used to estimate the amount of time needed for a cloud of interstellar gas to collapse into a proto-star, where d is the initial radius of the cloud and M is the cloud's total mass.

    Check my answer,

    t(r) = √{d/[2G(M+m)]} {√(rd−r²) + d Arctan[√(d/r−1)]}
     
  2. jcsd
  3. May 29, 2016 #2

    Simon Bridge

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    To check - try another approach:
    ie. figure out the equation if ##M>>m## and ##\delta r \to 0## ... see if you get the uniform gravity (time to fall h).

    Since they are initially at rest, ##M## and ##m## just fall directly towards each other ... so this is a 1D problem: choose coordinates so the x axis connects both centers of mass and put ##M## at ##x_M## and ##m## at ##x_m## so that ##x_M>x_m##, then, for ##\dot x_M,\dot x_m <\!< c## they should obey Newton's laws as:

    ##-Gm = (x_M-x_m)^2\ddot x_M##
    ##GM = (x_M-x_m)^2\ddot x_m##
    ... using the +x direction for +ve on vectors, and the direction is indicated by the sign.

    Change variable to ##x=x_M-x_m## (the separation of the masses) we can combine these equations giving the IVP:
    ##G(M+m) = x^2\ddot x : x(0)=d, \dot x(0)=0##
    ... which is awkward sure, but not impossible.
    You can solve this to see if it looks like your result... or just see if this can be made to look like your initial DE.

    The vis-viva approach is basically conservation of energy... so how about trying without using the official vis-viva equation?
    $$mv_m^2 + Mv_M^2 = 2GMm\left(\frac{1}{x}-\frac{1}{d}\right)$$ ... defining (by conservation of momentum): ##mv_m - Mv_M = 0##

    ...but you will prefer to use the speed at which the separation is closing instead: ##v=-\dot x = v_M+v_m##
    manipulate the simultaneous equations to get a DE.

    It seems you can get an equation for v(x) fairly easily... and it is easier to solve.
    Is there a way to use this to get the time as a function of x as t(x)?
     
  4. May 29, 2016 #3
    You can check my result numerically. For example, suppose that Earth halted in its orbit around the sun, so that initially (at time t=0) Earth and Sun were at rest and separated by d=1.495978707e11 meters. Find the time until Earth falls through the sun's photosphere (r=6.96e8 m).

    GM = 1.3271284e20
    dt = 0.01 (one-hundredth second increments in time)
    t = 0
    v = 0
    r = 1.4959787e11
    Repeat
    t = t + dt
    a = −GM/r²
    r = r + v dt + ½ a (dt)²
    v = v + a dt
    Until r ≤ 6.96e8

    When you run that, you find out that the elapsed time is 5577992.68 seconds.

    Now, solving the equation:

    t(r) = √{d/[2G(M+m)]} {√(rd−r²) + d Arctan[√(d/r−1)]}

    where
    d = 1.4959787e11 m
    r = 6.96e8 m
    G(M+m) = 1.3271284e20 m³ sec⁻²

    ...the result is:

    t = 5577992.74 sec

    From the limiting equation

    t = π √[d³/(8GM)]

    we find that the time for Earth to fall from 1 AU to the center of the sun would be

    t = 5578753.56 sec

    So, neglecting forces that dissipate kinetic energy and regarding the sun as a point mass, it would take only 760.82 seconds for Earth to move from the sun's photosphere to its core.
     
    Last edited: May 29, 2016
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