### Velocity Transformation into the Sphere of Influence of Jupiter

Larry Bogan

#### velocity

The path of the spacecraft within the Jupiter's sphere of influence (SoI) depends on the velocity and position of the spacecraft as it enters the sphere. The velocity is set by the spacecrafts orbit and the velocity of Jupiter. When translated to the reference frame of Jupiter, the velocity of the space craft is 7.91 km/s. This is much greater than the escape velocity from Jupiter and the spacecraft will zoom past the planet and leave the sphere of influence with the same velocity. Only the direction will be changed.

### Escape velocity:

vesc = [2MJG/r]1/2
MJG = 1.267x108 km (km/s)2 = 0.847 AU(km/s)2
r = 0.322 AU (radius of sphere of influence)
vesc = [2x0.847 AU(km/s)2/0.322 AU]1/2 = 2.29 km/s

The other aspect of the passage of Jupiter depends on the spacecrafts position relative to the planet. This depends on the timing. Since the angle of the velocity is -45o then if the spacecraft enters the SoI at forward by this angle, it will be aimed directly at the planet. This position is adjusted by the timing of arrival. I have chosed an angle of 40o and so the craft misses the planet by 4.86o. The impact parameter that this corresponds to is determined by geometry and equals 0.273 AU or 4.085 million km. b = 0.322 AU sin(44.9o - 40o) = 0.0273 AU

#### hyperbolic orbit

The angle of deflection can now be determined using
cot(q) = b v2/(GMJ)
q = 26.4o and total deflection = 2q = 52.8o
Since the orbit is symmetric about Jupiter, the angle that the spacecraft leave the SoI of Jupiter is determined. As mention above, the magnitude of this velocity is the same as that with which the spacecraft entered. This is all that is necessary to determine where the new orbit for the spacecraft (We will show that result on the next page).

It is interesting to see the path of the spacecraft near Jupiter. The parameters of that hyperbolic orbit are:

• e = 2.25
• a = 0.0148 AU = 2.21 million km
• rp = 2.76 million km
Note that the craft says well outside the orbits of the Galilean Satellites. This is different than the flight of Voyager 1 and Voyager II.