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 = 4100000 Msol
Schwarzschild radius Rs = 0.080964578463818 a.u. (12112128.5646km)
Phase 1: emitter moves towards event horizon
Emission distance R2 = Rs + d (variable)
Observer distance R1 = 2.500000e+20m
Emitted frequency f2 = 1.0E+20Hz
d [m] f1 [Hz] z
3.000000e+20 1.000000e+20 -4.037509e-12 (-0.00)
3.000000e+19 1.000000e+20 1.776445e-10 (0.00)
3.000000e+18 1.000000e+20 1.994464e-9 (0.00)
3.000000e+17 1.000000e+20 2.016266e-8 (0.00)
3.000000e+16 9.999998e+19 2.018446e-7 (0.00)
3.000000e+15 9.999980e+19 2.018662e-6 (0.00)
3.000000e+14 9.999798e+19 2.018665e-5 (0.00)
3.000000e+13 9.997982e+19 2.018484e-4 (0.00)
3.000000e+12 9.979874e+19 2.016655e-3 (0.00)
3.000000e+11 9.804045e+19 1.998714e-2 (0.02)
3.000000e+10 8.440283e+19 1.847943e-1 (0.18)
3.000000e+9 4.455514e+19 1.244410e+0 (1.24)
3.000000e+8 1.554667e+19 5.432244e+0 (5.43)
3.000000e+7 4.970650e+18 1.911809e+1 (19.12)
3.000000e+6 1.573608e+18 6.254822e+1 (62.55)
3.000000e+5 4.976741e+17 1.999347e+2 (199.93)
3.000000e+4 1.573801e+17 6.344043e+2 (634.40)
3.000000e+3 4.976802e+16 2.008323e+3 (2008.32)
3.000000e+2 1.573803e+16 6.353035e+3 (6353.04)
3.000000e+1 4.976802e+15 2.009222e+4 (20092.22)
3.000000e+0 1.573803e+15 6.353935e+4 (63539.35)
3.000000e-1 4.976802e+14 2.009312e+5 (200931.23)
3.000000e-2 1.573803e+14 6.354025e+5 (635402.51)
3.000000e-3 4.976802e+13 2.009321e+6 (2009321.32)
3.000000e-4 1.573803e+13 6.354034e+6 (6354034.09)
3.000000e-5 4.976802e+12 2.009322e+7 (20093222.21)
3.000000e-6 1.573803e+12 6.354035e+7 (63540349.86)
3.000000e-7 4.976802e+11 2.009322e+8 (200932231.06)
3.000000e-8 1.573803e+11 6.354035e+8 (635403507.65)
3.000000e-9 4.976802e+10 2.009322e+9 (2009322319.59)
3.000000e-10 1.573803e+10 6.354035e+9 (6354035085.47)
3.000000e-11 4.976802e+9 2.009322e+10 (20093223204.86)
3.000000e-12 1.573803e+9 6.354035e+10 (63540350863.66)
3.000000e-13 4.976802e+8 2.009322e+11 (200932232057.58)
3.000000e-14 1.573803e+8 6.354035e+11 (635403508645.61)
3.000000e-15 4.976802e+7 2.009322e+12 (2009322320584.78)
3.000000e-16 1.573803e+7 6.354035e+12 (6354035086465.10)
3.000000e-17 4.976802e+6 2.009322e+13 (20093223205856.87)
3.000000e-18 1.573803e+6 6.354035e+13 (63540350864659.93)
3.000000e-19 4.976802e+5 2.009322e+14 (200932232058578.69)
3.000000e-20 1.573803e+5 6.354035e+14 (635403508646612.00)
3.000000e-21 4.976802e+4 2.009322e+15 (2009322320585786.75)
3.000000e-22 1.573803e+4 6.354035e+15 (6354035086466120.00)
3.000000e-23 4.976802e+3 2.009322e+16 (20093223205857868.00)
3.000000e-24 1.573803e+3 6.354035e+16 (63540350864661208.00)
3.000000e-25 4.976802e+2 2.009322e+17 (200932232058578656.00)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 1.000000e+12m
Observer distance R1 = variable
Emitted frequency f2 = 1.0E+20Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 1.093946e+25 -1.00
1.000000e+1 3.459361e+24 -1.00
1.000000e+2 1.093946e+24 -1.00
1.000000e+3 3.459361e+23 -1.00
1.000000e+4 1.093947e+23 -1.00
1.000000e+5 3.459376e+22 -1.00
1.000000e+6 1.093991e+22 -0.99
1.000000e+7 3.460789e+21 -0.97
1.000000e+8 1.098453e+21 -0.91
1.000000e+9 3.599335e+20 -0.72
1.000000e+10 1.478090e+20 -0.32
1.000000e+11 1.052475e+20 -0.05
1.000000e+12 1.000000e+20 0.00
1.000000e+13 9.946002e+19 0.01
1.000000e+14 9.940586e+19 0.01
1.000000e+15 9.940044e+19 0.01
1.000000e+16 9.939990e+19 0.01
1.000000e+17 9.939985e+19 0.01
1.000000e+18 9.939984e+19 0.01
1.000000e+19 9.939984e+19 0.01
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.