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Delta-v for Hohmann transfer from hyperbolic trajectory to circular orbit

  1. Aug 18, 2011 #1
    I get different result than stated in the book.
    What am I doing wrong?

    1. The problem statement, all variables and given/known data

    A spacecraft returning from a lunar mission approaches earth on a hyperbolic trajectory.
    At its closest approach A it is at an altitude of 5000 km, traveling at 10 km/s. At
    A retrorockets are fired to lower the spacecraft into a 500 km altitude circular orbit,
    where it is to rendezvous with a space station.
    Verify that the total delta-v required to lower the spacecraft from the hyperbola into the parking orbit is 6.415 km/s.

    rEarth = 6378
    Gravitational parameter μ = 398600

    2. Relevant equations

    r - radius
    e - eccentricity
    A - apogee
    P - perigee

    h - angular momentum
    v - velocity

    r = altitude + rEarth
    e = (rA - rP) / (rA + rP)
    rP = (h2/μ)*(1/(1+e))
    vA = h/rA
    vP = h/rP
    vcircular = sqrt(μ/r)

    3. The attempt at a solution

    I get h = 58458,

    Speed at apogee of the transfer orbit:
    vA = 5.1378 km/s,

    Delta-v at apogee:
    ΔvA = 10-5.1378 = 4.86219 km/s

    Speed at perigee of the transfer orbit:
    vP = 58458/6878 = 8.499 km/s

    Speed of the final orbit:
    vcircular = 7.6127 km/h

    Delta-v at perigee:
    ΔvP = 8.499 - 7.6127 = 0.8866 km/s

    Total delta-v:
    ΔvT = 4.86219 + 0.8866 = 5.749 km/s
     
    Last edited: Aug 18, 2011
  2. jcsd
  3. Aug 18, 2011 #2

    gneill

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    Staff: Mentor

    Hi Lujz, welcome to Physics Forums.

    I don't see anything wrong in your calculations. Is there perhaps an accompanying diagram that might introduce some "quirk" of the setup that is not included in the problem statement? An orbital plane change perhaps?
     
  4. Aug 18, 2011 #3
    Hi gneill,

    The accompanying diagram is this:
    [PLAIN]http://www.shrani.si/f/1l/r7/acGDhup/2/example62.png [Broken]

    The original question is: "Find the location of the space station
    at retrofire so that rendezvous will occur at B."
    It then proceeds with calculations for periods and the angle in question.
    Nothing I can notice that would affect total Δv.
     
    Last edited by a moderator: May 5, 2017
  5. Aug 18, 2011 #4

    gneill

    User Avatar

    Staff: Mentor

    Okay, so I don't see anything there that would affect your solution method. I suppose that the text's proposed answer is in error.
     
  6. Aug 18, 2011 #5

    D H

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    Staff Emeritus
    Science Advisor

    Last edited: Aug 18, 2011
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