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 = 565000000000000.000000Hz
d [m] f1 [Hz] z
1.000000e+12 5.650000e+14 -8.396640e-9 (-0.00)
1.000000e+11 5.650000e+14 4.897159e-9 (0.00)
1.000000e+10 5.649999e+14 1.378351e-7 (0.00)
1.000000e+9 5.649992e+14 1.467214e-6 (0.00)
1.000000e+8 5.649917e+14 1.476091e-5 (0.00)
1.000000e+7 5.649166e+14 1.476881e-4 (0.00)
1.000000e+6 5.641673e+14 1.475990e-3 (0.00)
1.000000e+5 5.568349e+14 1.466337e-2 (0.01)
1.000000e+4 4.964134e+14 1.381642e-1 (0.14)
1.000000e+3 2.841321e+14 9.885114e-1 (0.99)
1.000000e+2 1.022354e+14 4.526461e+0 (4.53)
1.000000e+1 3.281681e+13 1.621679e+1 (16.22)
1.000000e+0 1.039338e+13 5.336155e+1 (53.36)
1.000000e-1 3.287175e+12 1.708801e+2 (170.88)
1.000000e-2 1.039512e+12 5.425244e+2 (542.52)
1.000000e-3 3.287230e+11 1.717772e+3 (1717.77)
1.000000e-4 1.039514e+11 5.434235e+3 (5434.23)
1.000000e-5 3.287230e+10 1.718672e+4 (17186.72)
1.000000e-6 1.039514e+10 5.435135e+4 (54351.35)
1.000000e-7 3.287230e+9 1.718762e+5 (171876.21)
1.000000e-8 1.039514e+9 5.435225e+5 (543522.47)
1.000000e-9 3.287230e+8 1.718771e+6 (1718771.13)
1.000000e-10 1.039514e+8 5.435234e+6 (5435233.71)
1.000000e-11 3.287230e+7 1.718772e+7 (17187720.32)
1.000000e-12 1.039514e+7 5.435235e+7 (54352346.15)
1.000000e-13 3.287230e+6 1.718772e+8 (171877212.16)
1.000000e-14 1.039514e+6 5.435235e+8 (543523470.47)
1.000000e-15 3.287230e+5 1.718772e+9 (1718772130.59)
1.000000e-16 1.039514e+5 5.435235e+9 (5435234713.66)
1.000000e-17 3.287230e+4 1.718772e+10 (17187721314.94)
1.000000e-18 1.039514e+4 5.435235e+10 (54352347145.61)
1.000000e-19 3.287230e+3 1.718772e+11 (171877213158.43)
1.000000e-20 1.039514e+3 5.435235e+11 (543523471465.07)
1.000000e-21 3.287230e+2 1.718772e+12 (1718772131593.33)
1.000000e-22 1.039514e+2 5.435235e+12 (5435234714659.78)
1.000000e-23 3.287230e+1 1.718772e+13 (17187721315942.40)
1.000000e-24 1.039514e+1 5.435235e+13 (54352347146606.43)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 6.957000e+8m
Observer distance R1 = variable
Emitted frequency f2 = 565000000000000.000000Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 3.071421e+16 -0.98
1.000000e+1 9.727464e+15 -0.94
1.000000e+2 3.122444e+15 -0.82
1.000000e+3 1.123507e+15 -0.50
1.000000e+4 6.430614e+14 -0.12
1.000000e+5 5.732836e+14 -0.01
1.000000e+6 5.658327e+14 -0.00
1.000000e+7 5.650822e+14 -0.00
1.000000e+8 5.650071e+14 -0.00
1.000000e+9 5.649996e+14 0.00
1.000000e+10 5.649989e+14 0.00
1.000000e+11 5.649988e+14 0.00
1.000000e+12 5.649988e+14 0.00
1.000000e+13 5.649988e+14 0.00
1.000000e+14 5.649988e+14 0.00
1.000000e+15 5.649988e+14 0.00
1.000000e+16 5.649988e+14 0.00
1.000000e+17 5.649988e+14 0.00
1.000000e+18 5.649988e+14 0.00
1.000000e+19 5.649988e+14 0.00
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.