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From: desj@brahms.BERKELEY.EDU (David desJardins)
Newsgroups: net.space
Subject: Re: Ulysses probe
Message-ID: <12181@ucbvax.BERKELEY.EDU>
Date: Wed, 5-Mar-86 01:08:40 EST
Article-I.D.: ucbvax.12181
Posted: Wed Mar  5 01:08:40 1986
Date-Received: Fri, 7-Mar-86 04:11:37 EST
References: <[MC.LCS.MIT.EDU].837830.860304.KFL>
Sender: usenet@ucbvax.BERKELEY.EDU
Reply-To: desj@brahms.UUCP (David desJardins)
Organization: University of California, Berkeley
Lines: 120
Summary: Detailed orbital mechanics analysis

In article <[MC.LCS.MIT.EDU].837830.860304.KFL> KFL@MC.LCS.MIT.EDU
("Keith F. Lynch") writes:
>
>Note that RJ/RE = (VJ/VE)**2 = (AE/AJ)**2 = 5.20

   Actually this should be RJ/RE = (VJ/VE)**2 = (AE/AJ)**.5 = 5.20.

>  To get VP and VA, I assumed that Ulysses will follow an elliptical
>path whose perihelion and starting point is on the orbit of Earth, and
>whose aphelion is on the orbit of Jupiter.  This is the least energy
>method of getting to Jupiter.

   This assumption is the problem.  If we relax this only slightly we
can achieve a much more satisfactory result.

>  ... Ulysses will approach Jupiter with a velocity of 7,450 meters per
>second.  But note that Jupiter is moving in the same direction at a
>velocity of 13,100 meters per second.  So Jupiter will actually
>overtake Ulysses.  The two will come together at a relative velocity
>of VJ-VA or 5,650 meters per second.
>  ... and will leave the vicinity of Jupiter at 5,650 meters per
>second, the same speed as it arrived.  Note, however that this is
>5,650 meters per second relative to Jupiter.  Not relative to the Sun.

   A very good analysis.  We desire to leave the Jupiter encounter with
an absolute (relative to Sun) velocity directly out of the ecliptic.
Its magnitude should be relatively small, depending on how close we wish
to pass to the solar pole.  Thus we need to leave with a relative (to
Jupiter) velocity slightly greater than that that of Jupiter (with a
component equal to Jupiter's velocity but in the opposite direction, and
a smaller component out of the plane of the ecliptic).
   Since I don't know how close Ulysses is supposed to pass to the Sun,
I am simply going to note that it should be much less than the orbital
velocity of Jupiter.  Thus we desire to leave Jupiter with a *relative*
velocity slightly greater than the orbital velocity of Jupiter; we must
arrive at Jupiter with this same relative velocity.

   I will use the following notation:

Vir = radial component of initial velocity (after leaving Earth)
Vit =  theta component of initial velocity
Vfr = radial component of final velocity   (when arriving at Jupiter)
Vft =  theta component of final velocity

   These quantities are related by the following equations:

                   RE * Vit = RJ * Vft                        (1)

1/2 (Vir^2 + Vit^2) - AE*RE = 1/2 (Vfr^2 + Vft^2) - AJ*RJ     (2)

   Here (1) represents conservation of angular momentum; (2) represents
conservation of energy.

   If we solve (1) and (2) with Keith Lynch's assumption that
Vir = Vfr = 0, we get his solution of:

>VP [ = Vit ] = Perihelion velocity of Ulysses = 3.86E+4  meters/second
>VA [ = Vft ] = Aphelion velocity of Ulysses   = 7.45E+3  meters/second

   But let us not make this assumption.  The *relative* velocity on
arrival at Jupiter is the vector (Vfr,Vft-VJ).  If we want its magnitude
to be approximately equal to VJ, we have the constraint equation:

       Vfr^2 + (Vft - VJ)^2 = VJ^2                            (3)

   We can now eliminate Vfr and Vft from (1), (2), and (3) with the
following result:

   1/2 Vir^2 + 1/2 Vit^2 - (RE/RJ)*VJ*Vit = AE*RE - AJ*RJ     (4)

   (4) is a constraint on the *initial* velocity which will cause
Ulysses to arrive at Jupiter with the correct *final* velocity.

   The relative velocity with which we leave the Earth is the vector
(Vir,Vit-VE).  In order to minimize the thrust requirement we can
minimize the magnitude of this vector:

                   Vir^2 + (Vit - VE)^2                       (5)

   Minimizing (5) with respect to the constraint (4), using Lagrange
multipliers, yields the solution:

                                              Vir = 0

     1/2 Vit^2 - (RE/RJ)*VJ*Vit - (AE*RE - AJ*RJ) = 0

   Using the quadratic formula with the actual values yields:

              Vir = 0         Vit = 4.05E+4 m/s

   Note that an initial velocity of 3.86E+4 m/s was required for the
minimum-energy path to Jupiter, so we see that a relatively small
addition to the initial delta-V will suffice to encounter Jupiter
with the required velocity.

>  The only explanation I can think of is that fuel is expended during
>the close pass to Jupiter.  This would have the effect of increasing
>the velocity with which Ulysses leaves Jupiter, i.e. increasing the
>radius of the sphere in the vector diagram.  Since a point directly
>over (or under) the origin is needed, the radius must be greater than
>13,100.  Thus a delta-vee (change of velocity) of 7,450 meters per
>second is necessary.

   We see that a much smaller delta-V will suffice if provided upon
leaving Earth orbit.

>  Can someone tell me how I can get technical information on these
>probes from JPL or NASA or wherever?  All I have been able to get is
>very nontechnical publications.

   Ditto.

   Note that my analysis seems to confirm that I was wrong when I
said that Ulysses would follow a hyperbolic path around the Sun and
leave the solar system.  It appears that additional launch energy
would be necessary to cause this to happen.  It should indeed go
into an eccentric orbit with perihelion near the site of its Jupiter
encounter.

   -- David desJardins