Photon redshift for Schwarzschild black holes
https://en.wikipedia.org/wiki/Gravitational_redshift
2 G M
Rs = -------
c^2
R1 (R2 - Rs)
f1 = f2 sqrt[ -------------- ]
R2 (R1 - Rs)
f2 - f1
z = ---------
f1
Where Rs = Schwarzschild radius
M = black hole mass
G = gravitational constant
c = speed of light in vacuum
f1 = observed frequency
R1 = observer distance from singularity (ie. center of mass of the black hole), > Rs
f2 = emitted frequency
R2 = emitter distance from singularity, > Rs
z = redshift
[-z = blueshift]
Calculated in two phases,
phase 1: Photon redshift from different emitter distances as observed at R1.
Emitter starts at 1e12 meters away from the black hole, and approaches
all the way to 1e-24m (billionth of a proton radius, or so).
For each emitter distance, observed frequency [Hz] and redshift [dimensionless] are displayed.
phase 2: Photon redshift from selected emitter distance at different observer distances.
Observer starts at 1 meter away from the event horizon, and gains distance
up the decades all the way to 1e18m (>100ly)
For each observer distance, observed frequency [Hz] and redshift [dimensionless] are displayed.
Input as optional GET parameters [with default]:
M = mass in Msols [1]
R1 = observer distance in meters [1.49597871e11]
f2 = emitted frequency in Hertz' [1e20]
d = emitter distance from event horizon in phase 2 in meters [0.001]
DIGITS = request more decimal places for calculations [128]
Examples:
Solar mass black hole observed from Earth: http://rubor.org/schwarzschildarb.php
Same but with an emitted visible photon: http://rubor.org/schwarzschildarb.php?f2=0.565e15
Sgr A* from 1au: http://rubor.org/schwarzschildarb.php?M=4.1e6
Gargantua from 3au (~Miller's planet): http://rubor.org/schwarzschildarb.php?M=1e8&R1=4.48e11
10 Sols & emit from 1m above EH in phase 2: http://rubor.org/schwarzschildarb.php?M=10&d=1
Msol, visible, emitted from Rsol: http://rubor.org/schwarzschildarb.php?f2=0.565e15&d=6.957e8
Solar mass observed from Venus, emit a microwave: http://rubor.org/schwarzschildarb.php?f2=2.45e9&R1=108208930000
ESA 360° black hole visualisation
Mass M = 10 Msol
Schwarzschild radius Rs = 1.9747458161907E-7 a.u. (29.541776986828km)
Phase 1: emitter moves towards event horizon
Emission distance R2 = Rs + d (variable)
Observer distance R1 = 1.495979e+11m
Emitted frequency f2 = 1.0E+20Hz
d [m] f1 [Hz] z
1.000000e+12 1.000000e+20 -8.396641e-8 (-0.00)
1.000000e+11 1.000000e+20 4.897156e-8 (0.00)
1.000000e+10 9.999986e+19 1.378350e-6 (0.00)
1.000000e+9 9.999853e+19 1.467204e-5 (0.00)
1.000000e+8 9.998524e+19 1.475992e-4 (0.00)
1.000000e+7 9.985263e+19 1.475901e-3 (0.00)
1.000000e+6 9.855486e+19 1.466328e-2 (0.01)
1.000000e+5 8.786079e+19 1.381641e-1 (0.14)
1.000000e+4 5.028888e+19 9.885112e-1 (0.99)
1.000000e+3 1.809476e+19 4.526461e+0 (4.53)
1.000000e+2 5.808285e+18 1.621679e+1 (16.22)
1.000000e+1 1.839536e+18 5.336154e+1 (53.36)
1.000000e+0 5.818009e+17 1.708801e+2 (170.88)
1.000000e-1 1.839844e+17 5.425243e+2 (542.52)
1.000000e-2 5.818106e+16 1.717772e+3 (1717.77)
1.000000e-3 1.839847e+16 5.434234e+3 (5434.23)
1.000000e-4 5.818107e+15 1.718672e+4 (17186.72)
1.000000e-5 1.839847e+15 5.435134e+4 (54351.34)
1.000000e-6 5.818107e+14 1.718762e+5 (171876.20)
1.000000e-7 1.839847e+14 5.435224e+5 (543522.42)
1.000000e-8 5.818107e+13 1.718771e+6 (1718770.98)
1.000000e-9 1.839847e+13 5.435233e+6 (5435233.23)
1.000000e-10 5.818107e+12 1.718772e+7 (17187718.79)
1.000000e-11 1.839847e+12 5.435234e+7 (54352341.32)
1.000000e-12 5.818107e+11 1.718772e+8 (171877196.89)
1.000000e-13 1.839847e+11 5.435234e+8 (543523422.17)
1.000000e-14 5.818107e+10 1.718772e+9 (1718771977.86)
1.000000e-15 1.839847e+10 5.435234e+9 (5435234230.67)
1.000000e-16 5.818107e+9 1.718772e+10 (17187719787.58)
1.000000e-17 1.839847e+9 5.435234e+10 (54352342315.66)
1.000000e-18 5.818107e+8 1.718772e+11 (171877197884.81)
1.000000e-19 1.839847e+8 5.435234e+11 (543523423165.64)
1.000000e-20 5.818107e+7 1.718772e+12 (1718771978857.11)
1.000000e-21 1.839847e+7 5.435234e+12 (5435234231665.41)
1.000000e-22 5.818107e+6 1.718772e+13 (17187719788580.05)
1.000000e-23 1.839847e+6 5.435234e+13 (54352342316662.84)
1.000000e-24 5.818107e+5 1.718772e+14 (171877197885809.09)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 1.000000e+0m
Observer distance R1 = variable
Emitted frequency f2 = 1.0E+20Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 1.000000e+20 0.00
1.000000e+1 3.162759e+19 2.16
1.000000e+2 1.001674e+19 8.98
1.000000e+3 3.215300e+18 30.10
1.000000e+4 1.156918e+18 85.44
1.000000e+5 6.621849e+17 150.02
1.000000e+6 5.903320e+17 168.40
1.000000e+7 5.826596e+17 170.63
1.000000e+8 5.818868e+17 170.85
1.000000e+9 5.818094e+17 170.88
1.000000e+10 5.818017e+17 170.88
1.000000e+11 5.818009e+17 170.88
1.000000e+12 5.818008e+17 170.88
1.000000e+13 5.818008e+17 170.88
1.000000e+14 5.818008e+17 170.88
1.000000e+15 5.818008e+17 170.88
1.000000e+16 5.818008e+17 170.88
1.000000e+17 5.818008e+17 170.88
1.000000e+18 5.818008e+17 170.88
1.000000e+19 5.818008e+17 170.88
Bonus GET parameters for controlling the loops:
p1s = phase 1 loop start, 'outerspace', f.e. 1e24
p1d = phase 1 loop step divider, f.e. 10 or 1e1 (must be > 1)
p1e = phase 1 loop end, ~event horizon, f.e. 1e-32
p2s = phase 2 loop start, ~event horizon, f.e. 1e-24
p2d = phase 2 loop step multiplier, f.e. 10 or 1e1
p2e = phase 2 loop end, 'outerspace', f.e. 1e32
Input is handled via PHPs sprintf, which limits
numerical range for mass/distance to about 53 decades;
the proton mass divided by solar mass is < 1e-57, and therefore
proton horizon is intractable (until I maybe fix it).
Example: http://rubor.org/schwarzschildarb.php?p1s=1e18&p1d=2&p1e=1e-32
Creative Commons Attribution by jussi.kantola @ gmail.com 2017
Visualizations by Professor Andrew Hamilton and used with permission.
No guarantees on accuracy or correctness of the results. source code.