One stage from LEO

Speed in AU/yr (only used for hypothetical delta-V), not end result

This section works from Au/yr

To confirm against New Horizons

To confirm against Voyager 1, set

back to required delta-V from

data, use a13-14 as 300,300k

a24 to 373k to get correct V at 4 AU.

Earth orbit, assuming no gravity

a24-27 as 720k, 100k, 85k, 380.

Jupiter to 5.285 AU, 12381 m/s

assist from Jupiter

These are just to get the C3 about

and 570 m/s (from Horizons).

right (2% off), but gets the velocities

Extra energy at Earth radius

at 2,3,4 AU just right (both tangential

Correct answer is 20870 m/s at 8 AU

E/kg (solar escape but not earth)

and radial). Difference is likely Earth's

non-circular orbit. Then plug in for

Jupiter (5.34, 12680, -509). These are

Radius of Earth (km)

the actual values for Jupiter on the

Perigee of Earth orbit (km)

date of the New Horizons flyby.

Apogee of Earth orbit (km)

Set the ISP to 0 for the Jupiter

perigee from center (m)

assist, making it unpowered.

apogee from center (m)

Finally, set the flyby distance to 32.4

Speed at perigee

Two limiting cases, double check: orbital speed (circular LEO orbit)

orbital speed (super highly eccentric orbit)

Then at the very end, you can compare

Final E/kg at perigee, assuming speed in A3

against the New Horizons numbers

Final V (solar system frame)

from the JPL trajectory.

delta v needed from orbit, if no other assists

52.09 AU, v=13838 m/s

Now work forward, this time including powered Jupiter gravity assist

payload mass (kg)

This section just to compute a conventional C3

delta v provided by stage

v earth frame, after Earth escape

v (Earth relative), m/s

Convert back to solar system coordinates

e/kg, after earth escape

V, m/s after Earth escape,earth relative

V m/s, solar frame

e/kg, solar frame

Table for comparison plot to JPL

e/kg after solar escape

Assuming no further assist

For checking against JPL

4 AU transverse velocity (m/s)

4 AU radial velocity (m/s)

Distance from sun (AU)

If you don't have specific values here (as from JPL Horizons),

Tangential speed of orbit around sun (m/s)

use 5.2 AU, sqrt(mu/r), and 0 (circular orbit)

Example of sqrt(MuSun/JupiterDist)

Angle of orbit to pure tangent

Not used; we assume Jupiter is just translating; coordinate systems are aligned

e/kg at jupiter distance

This section assumes we can do a passive gravity

v (solar sys coords) at Jupiter

turn at Jupiter. In the Jupiter frame,

v (along orbit), by conservation of angular momentum

incoming and outgoing velocities are

v (perpendicular to orbit)

equal. So find incoming velocity, then

speed, along orbit, in Jupiter frame

apply the same velocity in the direction

incoming velocity, Jupiter frame

of Jupiter's orbit. Does not check for feasibility.

outgoing velocity, solar frame, assuming turn is possible

e/kg at jupiter distance

au/year, passive gravity assist at Jupiter

These are theoretical best results. The required bend may end up

degrees bend, Jupiter frame

hitting Jupiter. See next section for more realistic calculations

Jupiter radius (km)

Flyby perijove in units of radius

Flyby distance (from center), m

Eqns from https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec16.pdf, page 6

turn angle (degrees)

incoming e/kg, Jupiter frame

speed at perijove

new e/kg after exiting Jupiter gravity well

exit speed, Jupiter frame

New turning angle

Total tuning angle (1/2 of first part, 1/2 of second)

entrance angle, jupiter frame

exit angle, jupiter frame

x comp jupiter frame

tangential v, solar frame

final v, solar frame

e/kg, solar frame

e/kg after solar escape

Table for comparison plot to JPL

For example with known results, plug in distance here

Plug in known value of velocity, e.g. from JPL Horizons