| Download
SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Project: BHLectures
Views: 20113Kernel: SageMath 9.3.beta7
Shadow and critical curve of a Kerr black hole
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
In [1]:
'SageMath version 9.3.beta7, Release Date: 2021-02-07'
In [2]:
Functions and for critical null geodesics
We use and denote simply by .
In [3]:
In [4]:
In [5]:
The radii and of the two horizons:
In [6]:
Critical radii , , , , and
In [7]:
In [8]:
In [9]:
In [10]:
We add the radius of the marginally stable orbit:
In [11]:
as well as the radius of outer and inner polar orbits:
In [12]:
In [13]:
In [14]:
In [15]:
In [16]:
In [17]:
In [18]:
In [19]:
In [20]:
In [21]:
In [22]:
In [23]:
In [24]:
In [25]:
In [26]:
In [27]:
rmin : 1.38628054232578 rmax : 3.95534727811920
rmin_col : 1.3862805284629751 rmax_col : 3.95534731767268
In [28]:
rmin : 1.54632082701639 rmax : 3.53769021424612
rmin_col : 1.3862805284629751 rmax_col : 3.95534731767268
In [29]:
rmin : 1.76846188276999 rmax : 3.23697774665377
rmin_col : 1.3862805284629751 rmax_col : 3.95534731767268
Case
In [30]:
rmin : 2.49118715973461 rmax : 2.49420141399888
rmin_col : 1.3862805284629751 rmax_col : 3.95534731767268
Shadow radius for the on-axis observer:
In [31]:
In [32]:
In [33]:
Extremal values:
In [34]:
Relative amplitude:
In [35]:
Shadow for
The partial shadow boundary:
In [36]:
rmin : 1.00000001000000 rmax : 3.99999996000000
rmin_col : 1 rmax_col : 4.0
With the NHEK line:
In [37]:
rmin : 1.00000001000000 rmax : 3.99999996000000
rmin_col : 1 rmax_col : 4.0
As soon as , the shadow boundary is a smooth closed curve:
In [38]:
rmin : 1.01637428079421 rmax : 3.99991107028894
rmin_col : 1.01637427063047 rmax_col : 3.9999111102880467
Screen coordinates for
In [39]:
In [40]:
In [41]:
Special values for :
In [42]:
In [43]:
Special values for :
In [44]:
In [45]:
Critical inclination for the existence of the NHEK line:
In [46]:
In [47]:
In [48]:
In [49]:
rmin : 1.00000001000000 rmax : 3.59326048984047
rmin_col : 1 rmax_col : 4.0
In [50]:
rmin : 1.00000001000000 rmax : 3.67516782408563
rmin_col : 1 rmax_col : 4.0
In [51]:
rmin : 1.13415225868399 rmax : 3.50327928110302
rmin_col : 1 rmax_col : 4.0
In [52]:
rmin : 1.00000001000000 rmax : 3.79787701838380
rmin_col : 1 rmax_col : 4.0
In [53]:
Cardiod shape for
In [54]:
In [55]:
rmin : 1.00000001000000 rmax : 3.99999996000000
rmin_col : 1 rmax_col : 4.0
Radial polynomial for and
Generic form of :
In [56]:
Specialization to and :
In [57]:
In [58]:
In [59]:
In [60]:
Comparing the critical curves at the same inclination angle
In [61]:
rmin : 2.34729637880682 rmax : 3.53208885091707
rmin_col : 2.3472963553338606 rmax_col : 3.5320888862379562
rmin : 1.00000001000000 rmax : 3.99999996000000
rmin_col : 1 rmax_col : 4.0
In [62]:
rmin : 2.48552158277147 rmax : 3.33625142631788
rmin_col : 2.3472963553338606 rmax_col : 3.5320888862379562
rmin : 1.05826009412625 rmax : 3.55486581066046
rmin_col : 1 rmax_col : 4.0
In [63]:
rmin : 2.59373553480987 rmax : 3.20003297615050
rmin_col : 2.3472963553338606 rmax_col : 3.5320888862379562
rmin : 1.50000001500015 rmax : 3.23205077524837
rmin_col : 1 rmax_col : 4.0
In [64]:
rmin : 2.88260669342186 rmax : 2.88382889826327
rmin_col : 2.3472963553338606 rmax_col : 3.5320888862379562
rmin : 2.41250630313641 rmax : 2.41592046802216
rmin_col : 1 rmax_col : 4.0
Comparing the critical curves at the same value of
In [65]:
rmin : 1.00000001000000 rmax : 3.99999996000000
rmin_col : 1 rmax_col : 4.0
rmin : 1.50000001500015 rmax : 3.23205077524837
rmin_col : 1 rmax_col : 4.0
rmin : 2.41404287406783 rmax : 2.41438424713942
rmin_col : 1 rmax_col : 4.0
In [ ]: