From: mjessick@verizon.net   
      
   Scott Lowther wrote:   
   > I'm trying to figure out time-of flight for hyperbolic fast transits   
   > from Earth to Jupiter, and it's not going well. Can anybody point me   
   > towards either a table of such (with, say, varius departure V's at   
   > earth, and the time of flight to Jupiter orbit), or perhaps a   
   > spreadsheet or some such that does the same?   
   >   
   > Alternatively... if you wanted to get to Jupiter in 3 months, or six   
   > months, and you had an arbitrarily short thrust time (i.e. high thrust   
   > system), how fast would you have to go? And how fast would you be going   
   > at Jupiter?   
   >   
   > Elliptical Hohmann orbits are *so* much easier...   
   >   
      
   That's a good homework problem :)   
      
      
   Given the departure state you calculate the eccentricity and semi-major   
   axis. Knowing the Jupiter radius, you get the true anomaly   
   (phase angle of the transfer) at that radius from inverting the   
   radius equation:   
      
       r = p / (1 + e * cos(true_anomaly))   
      
       where p = h^2 / mu, h = angular momentum, mu = G * Msun   
      
   With the true anomaly you can get the time-of-flight from Kepler's   
   equation:   
      
   t = sqrt(a^3 / mu) (E - e Sin(E))   
   where E is the eccentric anomaly.   
   (Here I have assumed starting from periapsis to simplify,   
   but this is not required)   
      
   Cos(E) = (e + cos(true_anomaly) / (1 + e * cos(true_anomaly))   
      
   These are for elliptical orbits. There are similar   
   time-of-flight equations for parabolic and hyperbolic orbits   
   if required.   
      
      
   Given the semi-major axis (hence energy) and the Jupiter radius   
   you calculate the velocity at Jupiter, or do the vectors if   
   you need velocity relative to Jupiter.   
      
      
   Looks like you can get down to 0.5 months before needing to   
   go hyperbolic   
      
   - Matt   
      
   --- SoupGate-Win32 v1.05   
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