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
A black hole system to play in Kerbal Space Program -- aesthetic, not physical
Mass M = 1 Msol
Schwarzschild radius Rs = 1.9747458161907E-8 a.u. (2.9541776986828km)
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 -8.396640e-9 (-0.00)
1.000000e+11 1.000000e+20 4.897159e-9 (0.00)
1.000000e+10 9.999999e+19 1.378351e-7 (0.00)
1.000000e+9 9.999985e+19 1.467214e-6 (0.00)
1.000000e+8 9.999852e+19 1.476091e-5 (0.00)
1.000000e+7 9.998523e+19 1.476881e-4 (0.00)
1.000000e+6 9.985262e+19 1.475990e-3 (0.00)
1.000000e+5 9.855485e+19 1.466337e-2 (0.01)
1.000000e+4 8.786079e+19 1.381642e-1 (0.14)
1.000000e+3 5.028887e+19 9.885114e-1 (0.99)
1.000000e+2 1.809476e+19 4.526461e+0 (4.53)
1.000000e+1 5.808285e+18 1.621679e+1 (16.22)
1.000000e+0 1.839536e+18 5.336155e+1 (53.36)
1.000000e-1 5.818008e+17 1.708801e+2 (170.88)
1.000000e-2 1.839844e+17 5.425244e+2 (542.52)
1.000000e-3 5.818106e+16 1.717772e+3 (1717.77)
1.000000e-4 1.839847e+16 5.434235e+3 (5434.23)
1.000000e-5 5.818107e+15 1.718672e+4 (17186.72)
1.000000e-6 1.839847e+15 5.435135e+4 (54351.35)
1.000000e-7 5.818107e+14 1.718762e+5 (171876.21)
1.000000e-8 1.839847e+14 5.435225e+5 (543522.47)
1.000000e-9 5.818107e+13 1.718771e+6 (1718771.13)
1.000000e-10 1.839847e+13 5.435234e+6 (5435233.71)
1.000000e-11 5.818107e+12 1.718772e+7 (17187720.32)
1.000000e-12 1.839847e+12 5.435235e+7 (54352346.15)
1.000000e-13 5.818107e+11 1.718772e+8 (171877212.16)
1.000000e-14 1.839847e+11 5.435235e+8 (543523470.47)
1.000000e-15 5.818107e+10 1.718772e+9 (1718772130.59)
1.000000e-16 1.839847e+10 5.435235e+9 (5435234713.66)
1.000000e-17 5.818107e+9 1.718772e+10 (17187721314.94)
1.000000e-18 1.839847e+9 5.435235e+10 (54352347145.61)
1.000000e-19 5.818107e+8 1.718772e+11 (171877213158.43)
1.000000e-20 1.839847e+8 5.435235e+11 (543523471465.07)
1.000000e-21 5.818107e+7 1.718772e+12 (1718772131593.33)
1.000000e-22 1.839847e+7 5.435235e+12 (5435234714659.76)
1.000000e-23 5.818107e+6 1.718772e+13 (17187721315942.28)
1.000000e-24 1.839847e+6 5.435235e+13 (54352347146606.30)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 1.000000e-5m
Observer distance R1 = variable
Emitted frequency f2 = 1.0E+20Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 3.162813e+17 315.17
1.000000e+1 1.001691e+17 997.31
1.000000e+2 3.215354e+16 3109.08
1.000000e+3 1.156937e+16 8642.51
1.000000e+4 6.621961e+15 15100.27
1.000000e+5 5.903420e+15 16938.33
1.000000e+6 5.826694e+15 17161.39
1.000000e+7 5.818966e+15 17184.18
1.000000e+8 5.818193e+15 17186.47
1.000000e+9 5.818115e+15 17186.70
1.000000e+10 5.818108e+15 17186.72
1.000000e+11 5.818107e+15 17186.72
1.000000e+12 5.818107e+15 17186.72
1.000000e+13 5.818107e+15 17186.72
1.000000e+14 5.818107e+15 17186.72
1.000000e+15 5.818107e+15 17186.72
1.000000e+16 5.818107e+15 17186.72
1.000000e+17 5.818107e+15 17186.72
1.000000e+18 5.818107e+15 17186.72
1.000000e+19 5.818107e+15 17186.72
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.