Periapse/Apoapse Speed Calculation

It occurred to me that the specifics of the calculation in the last post may be useful to others who want to repeat the work or do it for other spacecraft/orbiting bodies. Now this won't be plug-and-play for everyone, as the calculation I'll show you requires that you know the periapse (closest distance), apoapse (furthest distance), and period or semimajor axis of the orbit.

Fortunately, for Helios 1 and 2, we have those here. We've also got some fellow Illini who've done the calculation for Helios 2 so I can check my math.

These are:

Helios 1 Aphelion = 0.985 AU
Helios 1 Perihelion = 0.309 AU
Helios 1 Period = 190 days = 16416000 seconds
Helios 2 Aphelion = 0.983 AU
Helios 2 Perihelion = 0.290 AU
Helios 2 Period = 187 days = 16156800 seconds

Now we need some equations:
That leaves µ, the standard gravitational parameter. Fortunately we have Wikipedia. For the sun, µ = 132,712,440,018 km³s^-2

Now plug this into Excel or you favorite calculator:

Helios 1 Semimajor Axis = 96759926 km
Helios 1 Eccentricity = 0.522
Helios 1 Aphelion Speed = 20.7 km/s = 46382 miles/hr
Helios 1 Perihelion Speed = 66.1 km/s = 147851 miles/hr

Helios 2 Semimajor Axis = 95738701 km
Helios 2 Eccentricity = 0.544
Helios 2 Aphelion Speed = 20.2 km/s = 45218 mph
Helios 2 Perihelion Speed = 68.5 km/s = 153273 mph

So these suckers were zoomin', even when furthest from the sun. These check well enough with the Aerospaceweb speed numbers. Any differences are from slight inconsistencies in our input parameters.