Summarium
Equations
Natural dimensionless units :
G
=
M
=
c
=
1
{\displaystyle {\rm {G=M=c=1}}}
Inner (-) and outer (+) ergospheres :
r
E
±
=
1
−
a
2
cos
2
θ
±
1
,
{
x
,
z
}
=
{
a
2
+
r
E
±
2
sin
θ
,
r
E
±
cos
θ
}
{\displaystyle {\rm {r_{E}^{\pm }={\sqrt {1-a^{2}\cos ^{2}\theta }}\pm 1,\ \{x,z\}=\{{\sqrt {a^{2}+{r_{E}^{\pm }}^{2}}}\sin \theta ,\ r_{E}^{\pm }\cos \theta \}}}}
Inner (-) and outer (+) horizons :
r
H
±
=
1
−
a
2
±
1
,
{
x
,
z
}
=
{
a
2
+
r
H
±
2
sin
θ
,
r
H
±
cos
θ
}
{\displaystyle {\rm {r_{H}^{\pm }={\sqrt {1-a^{2}}}\pm 1,\ \{x,z\}=\{{\sqrt {a^{2}+{r_{H}^{\pm }}^{2}}}\sin \theta ,\ r_{H}^{\pm }\cos \theta \}}}}
Shadow contours:
0
=
(
(
−
x
)
2
+
z
2
−
x
A
)
2
−
B
2
(
(
−
x
)
2
+
z
2
)
{\displaystyle {\rm {0=((-x)^{2}+z^{2}-x\ A)^{2}-B^{2}\ ((-x)^{2}+z^{2})}}}
Limaςon parameter series expansion :
α
=
{\displaystyle {\rm {\alpha =}}}
−
8892.68
a
10
+
30413.2
a
9
−
46107.4
a
8
+
{\displaystyle {\rm {-8892.68a^{10}+30413.2a^{9}-46107.4a^{8}+}}}
+
37064.7
a
7
−
18685.4
a
6
+
4666.5
a
5
−
3894.54
a
4
+
{\displaystyle {\rm {\ \ \ \ \ \ +37064.7a^{7}-18685.4a^{6}+4666.5a^{5}-3894.54a^{4}+}}}
+
49.5645
a
3
−
9672.25
a
2
+
2.27392
a
+
9669.01
a
tan
(
a
)
{\displaystyle {\rm {\ \ \ \ \ \ \ +49.5645a^{3}-9672.25a^{2}+2.27392a+9669.01a\ \tan(a)}}}
β
{\displaystyle {\rm {\beta }}}
=
5.19058
−
0.343743
a
tan
(
a
)
+
0.0284803
a
−
0.0470795
a
27.5224
tan
(
a
)
{\displaystyle {\rm {=5.19058-0.343743a\ \tan(a)+0.0284803a-0.0470795a^{27.5224}\tan(a)}}}
Fourier transformation for the observer's inclination angle θ:
A
=
α
sin
θ
+
a
sin
3
θ
cos
2
θ
/
5
{\displaystyle {\rm {A=\alpha \sin \theta +a\sin ^{3}\theta \cos ^{2}\theta /5}}}
B
=
β
+
0.23
cos
4
θ
(
1
−
1
−
a
4
)
{\displaystyle {\rm {B=\beta +0.23\cos ^{4}\theta \ (1-{\sqrt {1-a^{4}}})}}}
Code: Source (Mathematica Syntax)
References
↑ Andreas de Vries: Shadows of rotating black holes
↑ Hung-Yi Pu, Kiyun Yun, Ziri Younsi, Suk Jin Yoon: Odyssey
↑ Max Planck Institute: Scientists to image event horizon of black hole
↑ Claudio Paganini, Blazej Ruba, Marius Oancea: Null Geodesics on Kerr Spacetimes
↑ Naoki Tsukamoto: Kerr-Newman and rotating regular black hole shadows in flat spacetime
↑ Grenzebach, Perlick & Lämmerzahl: Photon Regions and Shadows of Kerr–Newman–NUT Black Holes
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Anglica Contours of the observed shadow and the invisible surfaces of a rotating black hole
Germanica Konturen des beobachteten Schattens und der unsichtbaren Flächen eines rotierenden schwarzen Lochs