Space Forum / Space Flight / December 2003

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

JRS:  In article <3FEFAAC4.1A31@ix.netcomARGH.com>, seen in
news:sci.space.tech, Scott Lowther <scottlowtherHATESSPAM@ix.netcomARGH.
com.retro.com> posted at Mon, 29 Dec 2003 04:17:09 :-

 Earth's orbit radius ~ 150 Gm
Jupiter's orbit radius ~ 750 Gm
Distance ~ 600 - 900 Gm
Take worst case ; 900 Gm in 90 days is 10Gm / day.
One day is 0.1 Ms (-15%)
Hence speed required is 0.1 Mm/s

That's reasonable, since IIRC Ulysses took something like 15 months,
implying 20km/s, which seems feasible.

As you described it, you would be going 0.1 Mm/s if you did not want to
stop, or were using Jovo-braking - and, I suggest, nearer 0.0 Mm/s if
you did want to stop gently.

Earth's speed is about 30 km/s - so a significant navigational
correction hardly affecting the principle.