now what I want to condense into a post or so is a sampling of how the formulas look, which this thread has been illustrating
1. for a satellite in circular orbit
mass = 4 x period x speed
3
e.g. a planet's year is E50 and its speed is E-4 (both very like Earth's)
how massive is its star?
4 E(50-12) = 4E38
e.g. a planet's mass is E33, its year is E50 and the speed of a synchronous satellite circling it is E-5 (similar to Earth case as well)
how many of this planet's days to a year?
4 period E-15 = E33, 4 period =E48, 400
days in a year.
e.g. you are orbiting a small planet at the speed of a run, 6.7 mph, and find that full circuit takes 1 and 7/8 hours. What is the planet's mass?
speed = E-8, 4 x period = 450 minutes = E47, E47 E-24 = E23
2. for black hole radius, area, temperature, evaporation time
radius = (1/4pi) mass
area = (1/4pi) mass
2
temp = mass
-1
evaporation time = (80/pi) mass
3
3. radiant energy density and brightness
(energy per unit volume, power per unit area)
energy density = (pi
2/15) temp
4
brightness = (pi
2/60) temp
4
4. average photon energy
3zeta(4)/zeta(3) = 2.701 tells the average thermal photon energy at some temp. Multiply the temperature by 2.701.
avg photon energy = 2.701 temp
Since sun surface temp is 2E-28, the average sunlight photon has energy 5.402E-28.
Sun core temp is 5E-25, so the average core photon has energy 13.5E-25.
Room temperature is 1.04E-29, so the average energy of a photon in the room with you right now is 2.8E-29
5. critical density of universe
(just multiply the square of the Hubble parameter by 3)
H = (5/8)E-60
H
2 = (25/64)E-120
critical density = 3(25/64)E-120 = (75/64)E-120
It's the overall concentration of energy needed in the universe so that it can be spatially flat---too little makes negative curvature and too much makes positive curvature, either way triangles don't add up to 180 degrees--- and since it looks flat, folks think the actual density is at or close to critical.
This is where "0.83 joules per cubic km" comes from. It is just a metric translation of 1.2E-120
6. radian time in low orbit.
(time to go one radian, that is 1/2pi of full circle, lowest possible orbit)
radiantime
2 = 6/density
e.g. if density of planet is E-91 (similar to water) then square of radiantime is 6E91 = 60E90, so radiantime roughly 8E45 = 8 x 4.5 minutes.
e.g. if density of planet is 6E-91 (similar to Earth) then square of radiantime is E91 = 10E90, so radiantime roughly 3E45 = 3 x 4.5 minutes.
7. the heat capacity of water, per molecule
For the liquid, it is 9
So making some liquid water's temperature increase by E-30 takes an amount of energy equal to (the number of molecules) x 9E-30. The latent heat of vaporization is 1.7E-28 per molecule.
for a metallic solid, heat capacity is about 3 per atom
for a biatomic gas like air, 5/2 per molecule at constant volume, 7/2 per molecule at constant pressure
8. some 1/137 stuff
1/137 (more exactly 1/137.036...) is the coulomb constant. it tells the force between two charges separated by a distance. just multiply the charges by 1/137 and divide by the square of the distance.
1/137 also tells the force between parallel currents (measured on a test segment with length equal half the separation). just multiply the currents by 1/137
(1/137)
2 tells the energy needed to ionize a hydrogen atom. multiply the rest energy of an electron (2.1E-22) by it and you get a quantity of energy called the Hartree----which is twice the ionization energy (so you still need to divide by two)
in each case i am assuming that the calculation is done in natural units terms, so that I don't have to specify the units each time I say something.