There is one answer to What spacecraft has had the greatest total propulsive delta-v? and I can not understand how its numbers have been calculated. Response to comments does not seem forthcoming so I'll ask separately for a good, clear, science and math-based explanation how this works. As we are told in school, please show all work!
From this answer:
Taking all of this in to account, the delta-v of each space craft defined as spacecraft only delta-v + $\sqrt{{v_E}^2 + C_3}$, where ${v_E}^2 = 11.19 km/s$, the escape velocity from Earth.. The latter part converts the $C_3$ to the effective delta-v, when taking in to account losses from atmospheric drag, gravity drag, ineffective trajectories, etc. This seems to be the fairest way to calculate the effective delta-v. Taking all of this in to account, the following is the delta-v.
- Dawn- 22.89 km/s
- PSP- ~17.2 km/s
- New Horizons- 17.61 km/s
- Cassini- 15.69 km/s
- Juno- <14.5 km/s
The numbers changed from one edit to the next but have since stabilized.
Values for C3 and delta-v are scattered throughout the text, but if I understand correctly, if inserted in that equation result in those values.
I think they are meant to be geocentric C3 values rather than heliocentric (see this answer for examples of a heliocentric C3 and how to show one's work), and when quoted are actually the square roots of C3.
I can't understand the math;
- why velocities are added in quadrature
- why the units don't seem to work
- and how this produces the correct total propulsive delta-v for these spacecraft, either starting from Earth or from LEO.
Please explain in a clear, systematic way why this is the correct way to calculate total propulsive delta-v if it is, or how it should be done if it is not.