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 = 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+18 1.000000e+20 -9.873728e-9 (-0.00)
5.000000e+17 1.000000e+20 -9.873726e-9 (-0.00)
2.500000e+17 1.000000e+20 -9.873723e-9 (-0.00)
1.250000e+17 1.000000e+20 -9.873717e-9 (-0.00)
6.250000e+16 1.000000e+20 -9.873706e-9 (-0.00)
3.125000e+16 1.000000e+20 -9.873682e-9 (-0.00)
1.562500e+16 1.000000e+20 -9.873635e-9 (-0.00)
7.812500e+15 1.000000e+20 -9.873540e-9 (-0.00)
3.906250e+15 1.000000e+20 -9.873351e-9 (-0.00)
1.953125e+15 1.000000e+20 -9.872973e-9 (-0.00)
9.765625e+14 1.000000e+20 -9.872216e-9 (-0.00)
4.882812e+14 1.000000e+20 -9.870704e-9 (-0.00)
2.441406e+14 1.000000e+20 -9.867679e-9 (-0.00)
1.220703e+14 1.000000e+20 -9.861629e-9 (-0.00)
6.103516e+13 1.000000e+20 -9.849528e-9 (-0.00)
3.051758e+13 1.000000e+20 -9.825328e-9 (-0.00)
1.525879e+13 1.000000e+20 -9.776927e-9 (-0.00)
7.629395e+12 1.000000e+20 -9.680124e-9 (-0.00)
3.814697e+12 1.000000e+20 -9.486519e-9 (-0.00)
1.907349e+12 1.000000e+20 -9.099309e-9 (-0.00)
9.536743e+11 1.000000e+20 -8.324889e-9 (-0.00)
4.768372e+11 1.000000e+20 -6.776049e-9 (-0.00)
2.384186e+11 1.000000e+20 -3.678370e-9 (-0.00)
1.192093e+11 1.000000e+20 2.516990e-9 (0.00)
5.960464e+10 1.000000e+20 1.490771e-8 (0.00)
2.980232e+10 1.000000e+20 3.968915e-8 (0.00)
1.490116e+10 9.999999e+19 8.925202e-8 (0.00)
7.450581e+9 9.999998e+19 1.883778e-7 (0.00)
3.725290e+9 9.999996e+19 3.866292e-7 (0.00)
1.862645e+9 9.999992e+19 7.831320e-7 (0.00)
9.313226e+8 9.999984e+19 1.576137e-6 (0.00)
4.656613e+8 9.999968e+19 3.162145e-6 (0.00)
2.328306e+8 9.999937e+19 6.334154e-6 (0.00)
1.164153e+8 9.999873e+19 1.267814e-5 (0.00)
5.820766e+7 9.999746e+19 2.536600e-5 (0.00)
2.910383e+7 9.999493e+19 5.074122e-5 (0.00)
1.455192e+7 9.998985e+19 1.014897e-4 (0.00)
7.275958e+6 9.997971e+19 2.029791e-4 (0.00)
3.637979e+6 9.995942e+19 4.059268e-4 (0.00)
1.818989e+6 9.991890e+19 8.116989e-4 (0.00)
9.094947e+5 9.983799e+19 1.622750e-3 (0.00)
4.547474e+5 9.967676e+19 3.242885e-3 (0.00)
2.273737e+5 9.935663e+19 6.475331e-3 (0.01)
1.136868e+5 9.872552e+19 1.290928e-2 (0.01)
5.684342e+4 9.749857e+19 2.565609e-2 (0.03)
2.842171e+4 9.517592e+19 5.068590e-2 (0.05)
1.421085e+4 9.098877e+19 9.903674e-2 (0.10)
7.105427e+3 8.404360e+19 1.898586e-1 (0.19)
3.552714e+3 7.389130e+19 3.533392e-1 (0.35)
1.776357e+3 6.127876e+19 6.318867e-1 (0.63)
8.881784e+2 4.807854e+19 1.079930e+0 (1.08)
4.440892e+2 3.614984e+19 1.766264e+0 (1.77)
2.220446e+2 2.644020e+19 2.782120e+0 (2.78)
1.110223e+2 1.903162e+19 4.254414e+0 (4.25)
5.551115e+1 1.358092e+19 6.363269e+0 (6.36)
2.775558e+1 9.647753e+18 9.365108e+0 (9.37)
1.387779e+1 6.837922e+18 1.362433e+1 (13.62)
6.938894e+0 4.840803e+18 1.965773e+1 (19.66)
3.469447e+0 3.424971e+18 2.819732e+1 (28.20)
1.734723e+0 2.422531e+18 4.027914e+1 (40.28)
8.673617e-1 1.713240e+18 5.736896e+1 (57.37)
4.336809e-1 1.211532e+18 8.154011e+1 (81.54)
2.168404e-1 8.567141e+17 1.157251e+2 (115.73)
1.084202e-1 6.057994e+17 1.640711e+2 (164.07)
5.421011e-2 4.283688e+17 2.324437e+2 (232.44)
2.710505e-2 3.029039e+17 3.291377e+2 (329.14)
1.355253e-2 2.141859e+17 4.658842e+2 (465.88)
6.776264e-3 1.514525e+17 6.592732e+2 (659.27)
3.388132e-3 1.070931e+17 9.327667e+2 (932.77)
1.694066e-3 7.572630e+16 1.319545e+3 (1319.55)
8.470329e-4 5.354659e+16 1.866533e+3 (1866.53)
4.235165e-4 3.786316e+16 2.640090e+3 (2640.09)
2.117582e-4 2.677330e+16 3.734065e+3 (3734.06)
1.058791e-4 1.893158e+16 5.281179e+3 (5281.18)
5.293956e-5 1.338665e+16 7.469130e+3 (7469.13)
2.646978e-5 9.465790e+15 1.056336e+4 (10563.36)
1.323489e-5 6.693324e+15 1.493926e+4 (14939.26)
6.617445e-6 4.732895e+15 2.112772e+4 (21127.72)
3.308722e-6 3.346662e+15 2.987952e+4 (29879.52)
1.654361e-6 2.366447e+15 4.225644e+4 (42256.44)
8.271806e-7 1.673331e+15 5.976004e+4 (59760.04)
4.135903e-7 1.183224e+15 8.451387e+4 (84513.87)
2.067952e-7 8.366655e+14 1.195211e+5 (119521.08)
1.033976e-7 5.916119e+14 1.690287e+5 (169028.74)
5.169879e-8 4.183328e+14 2.390432e+5 (239043.15)
2.584939e-8 2.958059e+14 3.380585e+5 (338058.48)
1.292470e-8 2.091664e+14 4.780873e+5 (478087.30)
6.462349e-9 1.479030e+14 6.761180e+5 (676117.96)
3.231174e-9 1.045832e+14 9.561756e+5 (956175.60)
1.615587e-9 7.395148e+13 1.352237e+6 (1352236.92)
8.077936e-10 5.229160e+13 1.912352e+6 (1912352.21)
4.038968e-10 3.697574e+13 2.704475e+6 (2704474.84)
2.019484e-10 2.614580e+13 3.824705e+6 (3824705.41)
1.009742e-10 1.848787e+13 5.408951e+6 (5408950.68)
5.048710e-11 1.307290e+13 7.649412e+6 (7649411.82)
2.524355e-11 9.243935e+12 1.081790e+7 (10817902.36)
1.262177e-11 6.536449e+12 1.529882e+7 (15298824.65)
6.310887e-12 4.621968e+12 2.163581e+7 (21635805.72)
3.155444e-12 3.268225e+12 3.059765e+7 (30597650.29)
1.577722e-12 2.310984e+12 4.327161e+7 (43271612.44)
7.888609e-13 1.634112e+12 6.119530e+7 (61195301.59)
3.944305e-13 1.155492e+12 8.654323e+7 (86543225.87)
1.972152e-13 8.170562e+11 1.223906e+8 (122390604.18)
9.860761e-14 5.777460e+11 1.730865e+8 (173086452.75)
4.930381e-14 4.085281e+11 2.447812e+8 (244781209.36)
2.465190e-14 2.888730e+11 3.461729e+8 (346172906.50)
1.232595e-14 2.042640e+11 4.895624e+8 (489562419.71)
6.162976e-15 1.444365e+11 6.923458e+8 (692345814.00)
3.081488e-15 1.021320e+11 9.791248e+8 (979124840.42)
1.540744e-15 7.221825e+10 1.384692e+9 (1384691629.00)
7.703720e-16 5.106601e+10 1.958250e+9 (1958249681.85)
3.851860e-16 3.610912e+10 2.769383e+9 (2769383259.00)
1.925930e-16 2.553301e+10 3.916499e+9 (3916499364.70)
9.629650e-17 1.805456e+10 5.538767e+9 (5538766519.00)
4.814825e-17 1.276650e+10 7.832999e+9 (7832998730.40)
2.407412e-17 9.027281e+9 1.107753e+10 (11077533038.99)
1.203706e-17 6.383251e+9 1.566600e+10 (15665997461.80)
6.018531e-18 4.513640e+9 2.215507e+10 (22155066078.99)
3.009266e-18 3.191626e+9 3.133199e+10 (31331994924.59)
1.504633e-18 2.256820e+9 4.431013e+10 (44310132158.97)
7.523164e-19 1.595813e+9 6.266399e+10 (62663989850.18)
3.761582e-19 1.128410e+9 8.862026e+10 (88620264318.95)
1.880791e-19 7.979064e+8 1.253280e+11 (125327979701.36)
9.403955e-20 5.642050e+8 1.772405e+11 (177240528638.90)
4.701977e-20 3.989532e+8 2.506560e+11 (250655959403.72)
2.350989e-20 2.821025e+8 3.544811e+11 (354481057278.80)
1.175494e-20 1.994766e+8 5.013119e+11 (501311918808.44)
5.877472e-21 1.410513e+8 7.089621e+11 (708962114558.59)
2.938736e-21 9.973830e+7 1.002624e+12 (1002623837617.88)
1.469368e-21 7.052563e+7 1.417924e+12 (1417924229118.18)
7.346840e-22 4.986915e+7 2.005248e+12 (2005247675236.77)
3.673420e-22 3.526282e+7 2.835848e+12 (2835848458237.38)
1.836710e-22 2.493458e+7 4.010495e+12 (4010495350474.53)
9.183550e-23 1.763141e+7 5.671697e+12 (5671696916475.72)
4.591775e-23 1.246729e+7 8.020991e+12 (8020990700950.11)
2.295887e-23 8.815704e+6 1.134339e+13 (11343393832952.47)
1.147944e-23 6.233644e+6 1.604198e+13 (16041981401901.19)
5.739719e-24 4.407852e+6 2.268679e+13 (22686787665906.07)
2.869859e-24 3.116822e+6 3.208396e+13 (32083962803803.33)
1.434930e-24 2.203926e+6 4.537358e+13 (45373575331813.04)
7.174648e-25 1.558411e+6 6.416793e+13 (64167925607607.30)
3.587324e-25 1.101963e+6 9.074715e+13 (90747150663626.98)
1.793662e-25 7.792055e+5 1.283359e+14 (128335851215215.89)
8.968310e-26 5.509815e+5 1.814943e+14 (181494301327253.97)
4.484155e-26 3.896027e+5 2.566717e+14 (256671702430434.34)
2.242078e-26 2.754907e+5 3.629886e+14 (362988602654511.62)
1.121039e-26 1.948014e+5 5.133434e+14 (513343404860868.69)
5.605194e-27 1.377454e+5 7.259772e+14 (725977205309023.25)
2.802597e-27 9.740069e+4 1.026687e+15 (1026686809721737.38)
1.401298e-27 6.887269e+4 1.451954e+15 (1451954410618046.50)
7.006492e-28 4.870034e+4 2.053374e+15 (2053373619443474.75)
3.503246e-28 3.443634e+4 2.903909e+15 (2903908821236093.00)
1.751623e-28 2.435017e+4 4.106747e+15 (4106747238886949.50)
8.758115e-29 1.721817e+4 5.807818e+15 (5807817642472186.00)
4.379058e-29 1.217509e+4 8.213494e+15 (8213494477773899.00)
2.189529e-29 8.609086e+3 1.161564e+16 (11615635284944372.00)
1.094764e-29 6.087543e+3 1.642699e+16 (16426988955547798.00)
5.473822e-30 4.304543e+3 2.323127e+16 (23231270569888744.00)
2.736911e-30 3.043771e+3 3.285398e+16 (32853977911095596.00)
1.368456e-30 2.152271e+3 4.646254e+16 (46462541139777488.00)
6.842278e-31 1.521886e+3 6.570796e+16 (65707955822191192.00)
3.421139e-31 1.076136e+3 9.292508e+16 (92925082279554976.00)
1.710569e-31 7.609429e+2 1.314159e+17 (131415911644382384.00)
8.552847e-32 5.380679e+2 1.858502e+17 (185850164559109952.00)
4.276424e-32 3.804714e+2 2.628318e+17 (262831823288764768.00)
2.138212e-32 2.690339e+2 3.717003e+17 (371700329118219904.00)
1.069106e-32 1.902357e+2 5.256636e+17 (525663646577529536.00)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 1.000000e-3m
Observer distance R1 = variable
Emitted frequency f2 = 1.0E+20Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 3.162812e+18 30.62
1.000000e+1 1.001691e+18 98.83
1.000000e+2 3.215354e+17 310.01
1.000000e+3 1.156937e+17 863.35
1.000000e+4 6.621960e+16 1509.13
1.000000e+5 5.903419e+16 1692.93
1.000000e+6 5.826693e+16 1715.24
1.000000e+7 5.818965e+16 1717.52
1.000000e+8 5.818192e+16 1717.75
1.000000e+9 5.818114e+16 1717.77
1.000000e+10 5.818107e+16 1717.77
1.000000e+11 5.818106e+16 1717.77
1.000000e+12 5.818106e+16 1717.77
1.000000e+13 5.818106e+16 1717.77
1.000000e+14 5.818106e+16 1717.77
1.000000e+15 5.818106e+16 1717.77
1.000000e+16 5.818106e+16 1717.77
1.000000e+17 5.818106e+16 1717.77
1.000000e+18 5.818106e+16 1717.77
1.000000e+19 5.818106e+16 1717.77
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.