Below is a graph of Tisserand's Parameter for elliptical orbits (the eccentricity is limited to values from 0 to 0.99). Other limits that restrict the orbits are (1) having Perihelia less than the planet's orbital radius and (2) aphelia greater than the planet's orbital radius. These limits require solutions below the dashed black line on the graph.

Procedure for the example of a gravity assist to Neptune.

- 1. Determine what orbit is needed to get from Earth to Jupiter (Orbit #1)
- 2. Use Tisserand's Criteria to determine an orbit size and shape that will get the spacecraft from Jupiter to Neptune (Orbit #2)
- 3. Once these sizes and shapes are determined, the approach to Jupiter is fixed by the orbit velocities and angles near Jupiter.(Hyperbolic Orbit around Jupiter)

- Perihelion of orbit #1 at Earth = 1.0 AU
- Aphelion of orbit # 1 past Jupiter (a
_{J}= 5.20 AU) set = 6.0 AU - Calculate shape and size of orbit #1

a = (r_{per}+ r_{aph})/2 = 3.50 AU

e = (r_{aph}- r_{per})/a = 0.714

a/a_{J}= 0.673

Tisserand Parameter (TP) = a_{J}/a + 2 * sqrt(a/a_{J}*(1 - e^{2})) = 2.634 - Select semi-major axis (2) such that the aphelion is greater than the Neptune's orbital radius ( 30 AU).

Use the Tisserand Criteria (TP) to determine the eccentricity and aphelion.

Trial and error shows a = 18.00 AU is a good value.

e = sqrt(1 - a_{J}/a *(TP - a_{j}/a)^{2}/4)

e = 0.776

r_{aph}= a(1+e) = 31.97 AU

r_{per}= a(1-e) = 4.027 AU - Next use the properties of elliptical orbits to determine the velocities at Jupiter's orbit (r = a
_{j}= 5.20 AU)

v_{mean}= sqrt(k_{sun}/a)

where k_{sun}= G M_{sun}= 887.21 (km/s)^{2}AU

v = v_{mean}sqrt(2a/r - 1)

Results are:Orbit #1 Orbit #2 v _{mean}(km/s)15.92 7.02 v (km/s) 9.37 17.09 Angle from perihelion q(deg) 159.8 61.1 velocity wrt circular, f(deg) 36.8 26.3 - Next the velocities must be transformed to the velocities relative to the planet (Jupiter)

Jupiter's mean velocity (circular orbit) = sqrt(k_{sun}/a_{J}) = 13.06 km/s

Subtract this velocity and determine the components parallel and perpendicular to the motion of Jupiter.

v_{parallel}= v cos(f)- v_{J}

v_{perpendicular}= v sin(f)

v = sqrt(v_{par}^{2}+v_{per}^{2})

f = atan(v_{per}/v_{par})

The Results are:Orbit #1 Orbit #2 v _{par}(km/s)5.56 -2.25 v _{per}(km/s)5.61 7.57 v wrt Jupiter (km/s) 7.90 7.90 v angle (deg) 45.3 106.6 - Determine the approximate hyperbolic orbit past Jupiter

In the model that approximates the spacecraft-Jupiter interaction within a sphere-of-influence, the two velocities relative to Jupiter should be equal in magnitude and only different in direction. In order to estimate the hyperbolic orbit around Jupiter we will use the average of the two speeds.**Note:**

v_{c}(hyperbolic) ~ v_{avg}=(7.90+7.90)/2 = 7.90 km/s

Deflection Angle: d = 106.6 - 45.3 = 61.1 degrees

The asymptotic angle x = 90 + d/2 = 120.7 degrees

Eccentricity, e = -1/cos(x) = 1.96

semi-major axis = k_{jupiter}/v_{c}^{2}= - 0.014 AU ---- where k_{Jupiter}= GM_{Jupiter}= k_{Sun}/1047

b = a sqrt(e^{2}-1)= 0.23 AU = 3,420,00 km = impact parameter

r_{perijove}= a(e-1) = 0.013 AU = 1,910,000 km

Note: This is just outside the orbit of the outer of the four Galilean Satellites of Jupiter (Callisto) at 1,880,000 km. - This finishes the discussion of the orbits.

The following diagrams show the geometry of the orbits to scale.

and Flyby of Neptune.

The object of launching a spacecraft to Neptune's vicinity is to fly past it. Neptune must be at the orbit crossing location when the spacecraft gets there. The same is true for the interaction with Jupiter to get the gravity assist. The timing of the launch must be at a very special time when Jupiter will be the orbit cross of its orbit when the spacecraft gets there. To match of both the Neptune and Jupiter flybys would be a very difficult task to accomplish. But let's calculate the timing of the events.

- Procedure:
- The time of flight in an orbit is determined from Kepler's second law (Equal
areas are swept out by the radius vector from the Sun to the planet in equal times).
The angles from perihelion are used but must be converted into an angle called
the eccentric anamaly, which is sybolized by E.

cos(E) = (e+cos(n))/(1+e cos(n))

The time from Perihelion to get to the angle E is then given by

t - T = sqrt(a^{3}/k_{Sun})(E - e sin(E))

In this case it is best to use k_{Sun}= 39.47 AU^{3}/yr^{2} - We have all the angles we need except for the angle from perihelion
that the spacecraft encounters Neptunes orbit. This
Orbit #1 Orbit #2 Semi-major axis (AU) 3.50 18.0 Orbit Period (yrs) 6.55 76.37 Start Angle 0 deg 61.1 deg Finish Angle 159.8 deg 168.8 deg Time to Start (yrs) 0 1.23 Time to Finish (yrs) 1.87 26.81 Transit Time (yrs) 1.87 25.58

The time for the spacecraft to travel from Earth to Neptune is over 27 years. The times can be used to determine where the planets have to be at the time of launch for this whole scenario to occur. - Planet Positions using siderial periods:
Earth-to-Jupiter

1.87 yrsJupiter-to-Neptune

25.58 yrsEarth Motion (degrees) 673.7 9210.8 Jupiter Motion (degrees) 102.9 1406.4 Neptune Motion (degrees) 8.8 120.6 - Next, use these angles to determine the heliocentric longitudes of the planets at the
spacecraft launch time. (
*Note, I am using circular orbits and uniform motion of the planets for simplicity and as a result this is an approximation.*)

Earth Heliocentric Longitude: 0 degrees (reference)

Jupiter Heliocentric Longitude: 159.8 - 102.9 = 56.9 degrees

Neptune Heliocentric Longitude:159.8 - 61.1 + 168.8 - 8.8 - 120.6 = 138.1