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 = 1.082089e+11m
Emitted frequency f2 = 2450000000.000000Hz
d [m] f1 [Hz] z
1.000000e+12 2.450000e+9 -1.217325e-8 (-0.00)
1.000000e+11 2.450000e+9 1.120546e-9 (0.00)
1.000000e+10 2.450000e+9 1.340585e-7 (0.00)
1.000000e+9 2.449996e+9 1.463437e-6 (0.00)
1.000000e+8 2.449964e+9 1.475713e-5 (0.00)
1.000000e+7 2.449638e+9 1.476843e-4 (0.00)
1.000000e+6 2.446389e+9 1.475986e-3 (0.00)
1.000000e+5 2.414594e+9 1.466337e-2 (0.01)
1.000000e+4 2.152589e+9 1.381642e-1 (0.14)
1.000000e+3 1.232077e+9 9.885114e-1 (0.99)
1.000000e+2 4.433216e+8 4.526461e+0 (4.53)
1.000000e+1 1.423030e+8 1.621679e+1 (16.22)
1.000000e+0 4.506862e+7 5.336155e+1 (53.36)
1.000000e-1 1.425412e+7 1.708801e+2 (170.88)
1.000000e-2 4.507617e+6 5.425244e+2 (542.52)
1.000000e-3 1.425436e+6 1.717772e+3 (1717.77)
1.000000e-4 4.507625e+5 5.434235e+3 (5434.23)
1.000000e-5 1.425436e+5 1.718672e+4 (17186.72)
1.000000e-6 4.507625e+4 5.435135e+4 (54351.35)
1.000000e-7 1.425436e+4 1.718762e+5 (171876.21)
1.000000e-8 4.507625e+3 5.435225e+5 (543522.47)
1.000000e-9 1.425436e+3 1.718771e+6 (1718771.13)
1.000000e-10 4.507625e+2 5.435234e+6 (5435233.69)
1.000000e-11 1.425436e+2 1.718772e+7 (17187720.25)
1.000000e-12 4.507625e+1 5.435235e+7 (54352345.94)
1.000000e-13 1.425436e+1 1.718772e+8 (171877211.51)
1.000000e-14 4.507625e+0 5.435235e+8 (543523468.41)
1.000000e-15 1.425436e+0 1.718772e+9 (1718772124.10)
1.000000e-16 4.507625e-1 5.435235e+9 (5435234693.13)
1.000000e-17 1.425436e-1 1.718772e+10 (17187721250.03)
1.000000e-18 4.507625e-2 5.435235e+10 (54352346940.34)
1.000000e-19 1.425436e-2 1.718772e+11 (171877212509.32)
1.000000e-20 4.507625e-3 5.435235e+11 (543523469412.40)
1.000000e-21 1.425436e-3 1.718772e+12 (1718772125102.21)
1.000000e-22 4.507625e-4 5.435235e+12 (5435234694132.85)
1.000000e-23 1.425436e-4 1.718772e+13 (17187721251031.25)
1.000000e-24 4.507625e-5 5.435235e+13 (54352346941339.64)
Phase 2: observer moves away from the horizon
Emission distance R2 = Rs + 1.000000e-3m
Observer distance R1 = variable
Emitted frequency f2 = 2450000000.000000Hz
R1 = Rs + [m] f1 [Hz] z
1.000000e+0 7.748890e+7 30.62
1.000000e+1 2.454143e+7 98.83
1.000000e+2 7.877617e+6 310.01
1.000000e+3 2.834496e+6 863.35
1.000000e+4 1.622380e+6 1509.13
1.000000e+5 1.446338e+6 1692.93
1.000000e+6 1.427540e+6 1715.24
1.000000e+7 1.425646e+6 1717.52
1.000000e+8 1.425457e+6 1717.75
1.000000e+9 1.425438e+6 1717.77
1.000000e+10 1.425436e+6 1717.77
1.000000e+11 1.425436e+6 1717.77
1.000000e+12 1.425436e+6 1717.77
1.000000e+13 1.425436e+6 1717.77
1.000000e+14 1.425436e+6 1717.77
1.000000e+15 1.425436e+6 1717.77
1.000000e+16 1.425436e+6 1717.77
1.000000e+17 1.425436e+6 1717.77
1.000000e+18 1.425436e+6 1717.77
1.000000e+19 1.425436e+6 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.