The Orbital Factor table appears in MegaTraveller and The New Era, in World Tamer's Handbook, page 84. The numbers appear to be derived from what I can only assume is a formula. It has piqued my curiosity, but so far I've only come across a vague reference on a forum. Which is sort of surprising.
What is the formula behind or the logic involved in these numbers?
TL;DR: Given a distance in millions of kilometers: $$O = \frac{4574}{\sqrt{D}}$$
That's how you derive it. But for the logic behind it...
The formula from page 84 of the World Tamer's Handbook, $$ T=LOEG$$ takes four pieces of information generated from the RPG tables to calculate the fifth, temperature:
Now, outside of the Traveller RPG, in physics, the formula to calculate the radiative equilibrium temperature of a black body is:
$$T = \sqrt[4]{\frac{L(1-ɑ)}{16πσD^2}}$$
(Where σ is the Stefan–Boltzmann constant. This constant maps power per unit area - i.e., luminosity - to temperature. You can find out a lot more information about this formula on the astronomy stackexchange.)
Since we're looking to define O, we can simplify this formula by taking out other aspects of the Traveller formula.
Since the greenhouse effect G has a static effect which is not dependent in any way on L or O or E, we can remove G as effectively being a constant:
$$T = LOE$$
The effective albedo of the planet E is also a static value which is not dependent on L or O, so we can treat it as a constant as well:
$$T = LO$$
With those two terms removed, we can simplify the formula:
$$T = \sqrt[4]{\frac{L}{16πσD^2}}$$
We already know that the orbital factors value includes all the constant scalars to convert real-world physics values into game-table-friendly values, so we can remove the constants for the purpose of simplification:
$$T = \sqrt[4]{\frac{L}{D^2}}$$
Now, can we take this simplification, and use it to turn the orbital factors into a real-world measurement? Yes.
Take any orbital factor on the Orbital Factor table, and look up the distance in millions of kilometers on the Orbital Distances Table on page 195 of the TNE core rulebook. For illustration, let's use orbit 7:
$$D = 1495.9$$
Look up the orbital factor from page 84 of the World Tamer's rulebook.
$$O = 118.277$$
Multiply the orbital factor by the 4th root of square of the distance - which is to say, the square root of the distance. (In this case, the table is in millions of of kilometers.)
$$O \times \sqrt{D} = T$$
$$118.277 \times \sqrt{1495.9} = 4574.58$$
No matter what orbital factor you chose, you get within one of 4574. That is to say... the radiative equilibrium temperature in kelvin that a planet would be if it were some hypothetical ideal atmosphereless planet which was orbiting one million kilometers around a sun with a luminosity equal to Sol's.
