Atmospheric escape parametre statistics

[snorkack]

What could be most sensible parametre for atmospheric escape excluding higher order (and nonlinear!) effects?
Black body radiation is proportional to fourth power of temperature. Therefore temperature of black body is proportional to 4th root of incident radiation.
Radiation from a point source is proportional to inverse square of distance. Therefore temperature of a black body is proportional to inverse square root of distance to heat source.
Speed of same molecules is proportional to square root of temperature. Therefore speed of molecules heated by a black body is proportional to the inverse fourth root of distance to heat source.
Ratio of escape speed to the speed of molecules should therefore be proportional to escape speed multiplied by fourth root of distance to heat source.
Listing the planets and big satellites inside Solar System:
Earth: escape speed 11,2 km/s, distance to heat source 1 AU, escape parameter 11,2 km/s (has air)
Moon: escape speed 2,38 km/s, distance to heat source 1 AU, escape parameter 2,38 km/s (no atmosphere)
Mercury: here a correction is warranted. Escape of air while at perihelion is not compensated by non-escape at aphelion; and it is upper, low heat capacity air that is heated, which cannot store up heat in perihelion. So escape speed 4,25 km/s, distance to heat source 0,307 AU, escape parameter 3,16 km/s
Venus: escape speed 10,36 km/s, distance to heat source 0,718 AU, escape parameter 9,54 km/s (much gas but little water)
Mars: escape speed 5,03 km/s, distance to heat source 1,381 AU, escape parameter 5,45 km/s (some gas but little nitrogen)
Jupiter: escape speed 60,2 km/s, distance to heat source 4,95 AU, escape parameter 89,8 km/s
Io: escape speed 2,56 km/s, distance to heat source 4,95 AU, escape parameter 3,82 km/s
Europa: escape speed 2,02 km/s, distance to heat source 4,95 AU, escape parameter 3,01 km/s
Ganymedes: escape speed 2,74 km/s, distance to heat source 4,95 AU, escape parameter 4,09 km/s
Callisto: escape speed 2,44 km/s, distance to heat source 4,95 AU, escape parameter 3,64 km/s
Saturn: escape speed 36,1 km/s, distance to heat source 9,04 AU, escape parameter 62,6 km/s
Titan: escape speed 2,64 km/s, distance to heat source 9,04 AU, escape parameter 4,58 km/s (lots of nitrogen)
Uranus: escape speed 21,4 km/s, distance to heat source 18,3 AU, escape parameter 44,3 km/s
Neptune: escape speed 23,6 km/s, distance to heat source 29,8 AU, escape parameter 55,1 km/s
Triton: escape speed 1,45 km/s, distance to heat source 29,8 AU, escape parameter 3,39 km/s (observed 1...2 Pa atmosphere)
Pluto: escape speed 1,23 km/s, distance to heat source 29,6 AU, escape parameter 2,87 km/s (also observed 1 Pa atmosphere)
Eris: escape speed 1,38 km/s, distance to heat source 38,3 AU, escape parameter 3,43 km/s - atmosphere?

Now, what are planets and satellites outside Solar System for which both mass and radius (and therefore escape speed) are known by observation?
Note that when the heat source is other than sun, the ratio of escape speed multiplied by fourth root of distance to heat source must be further divided by eighth root of bolometric brightness of heat source.
I am suggesting that in the first order "linear" approximation - pure thermal escape from a pure black body - the rate and selectiveness of escape depends on a simple expression of 4 parametres:

1. body mass
2. body radius (1) and 2) combined give the escape speed)
3. body distance from heat source (note: at the closest where significantly variable, because escape at close approach is not compensated at distance!)
4. bolometric power of the heat source (3) and 4) combined give the black body temperature)

What goes beyond that is complications, like escape due to wind rather than heat, or colourful body temperature.
Note several anomalies in Solar System that may have been caused by the said higher order effects. To see them, reorder the entries by escape parametre:

1. Jupiter: escape speed 60,2 km/s, distance to heat source 4,95 AU, escape parameter 89,8 km/s
2. Saturn: escape speed 36,1 km/s, distance to heat source 9,04 AU, escape parameter 62,6 km/s
3. Neptune: escape speed 23,6 km/s, distance to heat source 29,8 AU, escape parameter 55,1 km/s
4. Uranus: escape speed 21,4 km/s, distance to heat source 18,3 AU, escape parameter 44,3 km/s
5. Earth: escape speed 11,2 km/s, distance to heat source 1 AU, escape parameter 11,2 km/s (has air)
6. Venus: escape speed 10,36 km/s, distance to heat source 0,718 AU, escape parameter 9,54 km/s (much gas but little water)
7. Mars: escape speed 5,03 km/s, distance to heat source 1,381 AU, escape parameter 5,45 km/s (some gas but little nitrogen)
8. Titan: escape speed 2,64 km/s, distance to heat source 9,04 AU, escape parameter 4,58 km/s (lots of nitrogen, and methane)
9. Ganymedes: escape speed 2,74 km/s, distance to heat source 4,95 AU, escape parameter 4,09 km/s
10. Io: escape speed 2,56 km/s, distance to heat source 4,95 AU, escape parameter 3,82 km/s
11. Callisto: escape speed 2,44 km/s, distance to heat source 4,95 AU, escape parameter 3,64 km/s
12. Eris: escape speed 1,38 km/s, distance to heat source 38,3 AU, escape parameter 3,43 km/s - atmosphere?
13. Triton: escape speed 1,45 km/s, distance to heat source 29,8 AU, escape parameter 3,39 km/s (observed 1...2 Pa atmosphere)
14. Mercury: escape speed 4,25 km/s, distance to heat source 0,307 AU, escape parameter 3,16 km/s
15. Europa: escape speed 2,02 km/s, distance to heat source 4,95 AU, escape parameter 3,01 km/s
16. Pluto: escape speed 1,23 km/s, distance to heat source 29,6 AU, escape parameter 2,87 km/s (also observed 1 Pa atmosphere)
17. Moon: escape speed 2,38 km/s, distance to heat source 1 AU, escape parameter 2,38 km/s (no atmosphere)

The anomalies from higher to lower escape parameters:
Titan holds to lots of nitrogen (150 000 Pa or so). Mars is short of nitrogen despite higher escape parameter. Why?
Titan holds much methane. Venus has little water. Even Mars has more water than Venus despite smaller escape parameter. Why?
Pluto and Triton hold 1...2 Pa nitrogen. Ganymedes, Io, Callisto, Mercury and Europa all have bigger escape parameters than Pluto, first three have bigger escape parameters than Triton, yet they lack nitrogen - Io has a few mPa of sulphur dioxide, the others far less. Why?
And two non-anomalies which nevertheless catch attention:
Vast gap between 11,2 km/s of Earth and 44,3 km/s of Uranus
Broad gap between 9,54 km/s of Venus and 5,45 km/s of Mars - flanked by anomalies. What is the anomaly - that Venus loses water, or that Titan does not lose methane?

 

[russ_watters]

Radiation from a point source is proportional to inverse square of distance. Therefore temperature of a black body is proportional to inverse square root of distance to heat source.

I don't think that follows and planets aren't point sources anyway.

Regardless, isn't this subject covered in textbooks or peer reviewed research? What do they have to say about atmospheric temperature distribution and gas escape?