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 = 1.495979e+11m
Emitted frequency f2 = 1.0E+20Hz
d [m] f1 [Hz] z
1.000000e+12 1.036859e+20 -3.554840e-2 (-0.04)
1.000000e+11 9.851618e+19 1.506166e-2 (0.02)
1.000000e+10 7.014853e+19 4.255465e-1 (0.43)
1.000000e+9 2.880695e+19 2.471385e+0 (2.47)
1.000000e+8 9.439265e+18 9.594045e+0 (9.59)
1.000000e+7 2.996018e+18 3.237764e+1 (32.38)
1.000000e+6 9.477760e+17 1.045102e+2 (104.51)
1.000000e+5 2.997242e+17 3.326400e+2 (332.64)
1.000000e+4 9.478147e+16 1.054058e+3 (1054.06)
1.000000e+3 2.997254e+16 3.335387e+3 (3335.39)
1.000000e+2 9.478151e+15 1.054958e+4 (10549.58)
1.000000e+1 2.997255e+15 3.336287e+4 (33362.87)
1.000000e+0 9.478151e+14 1.055048e+5 (105504.81)
1.000000e-1 2.997255e+14 3.336377e+5 (333637.65)
1.000000e-2 9.478151e+13 1.055057e+6 (1055057.06)
1.000000e-3 2.997255e+13 3.336386e+6 (3336385.55)
1.000000e-4 9.478151e+12 1.055058e+7 (10550579.64)
1.000000e-5 2.997255e+12 3.336386e+7 (33363864.46)
1.000000e-6 9.478151e+11 1.055058e+8 (105505805.39)
1.000000e-7 2.997255e+11 3.336387e+8 (333638653.55)
1.000000e-8 9.478151e+10 1.055058e+9 (1055058062.86)
1.000000e-9 2.997255e+10 3.336387e+9 (3336386544.51)
1.000000e-10 9.478151e+9 1.055058e+10 (10550580637.56)
1.000000e-11 2.997255e+9 3.336387e+10 (33363865454.11)
1.000000e-12 9.478151e+8 1.055058e+11 (105505806384.56)
1.000000e-13 2.997255e+8 3.336387e+11 (333638654550.10)
1.000000e-14 9.478151e+7 1.055058e+12 (1055058063854.59)
1.000000e-15 2.997255e+7 3.336387e+12 (3336386545510.06)
1.000000e-16 9.478151e+6 1.055058e+13 (10550580638555.02)
1.000000e-17 2.997255e+6 3.336387e+13 (33363865455109.56)
1.000000e-18 9.478151e+5 1.055058e+14 (105505806385558.59)
1.000000e-19 2.997255e+5 3.336387e+14 (333638654551105.62)
1.000000e-20 9.478151e+4 1.055058e+15 (1055058063855596.50)
1.000000e-21 2.997255e+4 3.336387e+15 (3336386545511056.00)
1.000000e-22 9.478151e+3 1.055058e+16 (10550580638555964.00)
1.000000e-23 2.997255e+3 3.336387e+16 (33363865455110560.00)
1.000000e-24 9.478151e+2 1.055058e+17 (105505806385559648.00)
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 3.162278e+18 30.62
1.000000e+1 1.000000e+18 99.00
1.000000e+2 3.162278e+17 315.23
1.000000e+3 1.000000e+17 999.00
1.000000e+4 3.162279e+16 3161.28
1.000000e+5 1.000004e+16 9998.96
1.000000e+6 3.162408e+15 31620.47
1.000000e+7 1.000413e+15 99957.74
1.000000e+8 3.175305e+14 314929.38
1.000000e+9 1.040462e+14 961110.20
1.000000e+10 4.272726e+13 2340425.25
1.000000e+11 3.042398e+13 3286879.67
1.000000e+12 2.890707e+13 3459360.29
1.000000e+13 2.875098e+13 3478141.60
1.000000e+14 2.873532e+13 3480036.60
1.000000e+15 2.873376e+13 3480226.27
1.000000e+16 2.873360e+13 3480245.24
1.000000e+17 2.873358e+13 3480247.14
1.000000e+18 2.873358e+13 3480247.33
1.000000e+19 2.873358e+13 3480247.35
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.