Visualizing a Binary Black Hole Merger
Author
Macy Maurer Levin
Title
Visualizing a Binary Black Hole Merger
Description
A project created for the Wolfram Summer Camp 2019 which models the the collision of two black holes.
Category
Essays, Posts & Presentations
Keywords
astrophysics, black holes, astronomy, physics, science, math, calculus, differential equations, general relativity, special relativity
URL
http://www.notebookarchive.org/2019-07-5k0auf5/
DOI
https://notebookarchive.org/2019-07-5k0auf5
Date Added
2019-07-12
Date Last Modified
2019-07-12
File Size
0.98 megabytes
Supplements
Rights
Redistribution rights reserved
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WOLFRAM SUMMER SCHOOL 2019
Visualising a Binary Black Hole Merger
Visualising a Binary Black Hole Merger
Macy Maurer Levin
Katja Della Libera and Jonathan Gorard
Introduction
Introduction
On September 14th, 2015, LIGO (the Laser Interferometer Gravitational-Wave Observatory), located in both Washington and Louisiana, detected a strange “ping” roughly 1.3 billion light years away from Earth. This “ping” was caused by gravitational waves as the result of a collision of two black holes which occurred 1.3 billion years ago just as Earth’s inner core began to form. The purpose of this project was to visualise the relationship of a binary black hole system and the eventual collision which marked a revolutionary event in the history of astrophysics.
Developing the Visualisation
Developing the Visualisation
In order to develop the visualisation, a set of differential equations had to be solved. Respectively, each equation modeled the average angular momentum magnitude along the Z-axis (seeing as the final result was to be three-dimensional), the rate of change of the semi-major axis, the eccentricity, the radius, and the angular velocity about the centre of inertia for the black holes--the result of which is show here:
A function of eccentricity:
A function of eccentricity:
In[]:=
e[t_]:=
1-*a[t]
(m+1)*(L[t])^2
2
m
In[]:=
m=1.1
Out[]=
1.1
Average angular momentum magnitude along the Z-axis and the rate of change of the semi-major axis :
Average angular momentum magnitude along the Z-axis and the rate of change of the semi-major axis :
In[]:=
system1=NDSolveL'[t]-1(1+(7/8)),a'[t]-435-366a[t]+37,L[0]4,a[0]100,{L,a},{t,0,10000}
32
7
2
1
2
m
1
2
(1+m)
5
5
1
7
2
a[t]
2
(1-)
2
e[t]
2
e[t]
2
8
m
1
2
a[t]
15
5
2
(1+m)
7
L[t]
(m+1)
2
L[t]
2
m
2
(1+m)
4
L[t]
4
m
2
a[t]
Out[]=
LInterpolatingFunction
,aInterpolatingFunction
 |  |
|
|  |
|
In[]:=
angularMomentum[t_]:=L[t]/.system1[[1]]
In[]:=
semiMajorAxis[t_]:=a[t]/.system1[[1]]
In[]:=
angularMomentum[1]
Out[]=
3.99998
In[]:=
semiMajorAxis[1]
Out[]=
99.9908
In[]:=
(a[t]/.system1)
Out[]=
InterpolatingFunction
[t]
| |
|
In[]:=
Plot[Evaluate[a[t]/.system1],{t,0,5}]
Out[]=
In[]:=
eccentricity[t_]:=
1-*semiMajorAxis[t]
(m+1)*(angularMomentum[t])^2
2
m
Radius:
Radius:
In[]:=
r[t_]:=
semiMajorAxis[t]*(1-)
2
(eccentricity[t])
1+eccentricity[t]Cos[θ[t]]
Angular velocity about the center of inertia:
Angular velocity about the center of inertia:
In[]:=
θ[t_]:=θ'[t]==
1(1+m)a[t](1-)
2
e[t]
2
r[t]
In[]:=
system3=NDSolveθ'[t],θ[0]0,θ,{t,0,10000}
1(1+m)*semiMajorAxis[t]*(1-)
2
eccentricity[t]
2
r[t]
After evaluating the differential equations, graphs could be determined from which the points representing the black holes could be plotted, a place from which the final visualization could be created. Initially, two rotating points were created using an animation on a circular plot orbiting parallel to each other around the center of mass, also known as a “barycenter”:
In[]:=
Animate[Show[{ContourPlot[x^2+1y^23,{x,-2,2},{y,-2,2}],Graphics[{RGBColor[0,0,0],Disk[{Sqrt[3]Sin[x],Sqrt[3]Cos[x]},0.1]},PlotRange{{-2,2},{-2,2}}],Graphics[{RGBColor[0,0,0],Disk[{Sqrt[3]Sin[(x+3)],Sqrt[3]Cos[(x+3)]},0.1]},PlotRange{{-2,2},{-2,2}}]}],{x,0,2Pi,0.01},SaveDefinitions"True"]
Out[]=
| ||
 |
Once this simple animation and the graphs were complete, the two were merged by a common function which allowed for the points plotted on the first animation to follow the path of the spiral graphs developed from the various initial conditions provided alongside the differential equations. Initially, the animation was developed via a function where one of the black holes remained stationary whilst the other one orbited around it following the path of the graph, gradually getting closer and closer as the distance decreased but the acceleration increased, a relationship modeled by the proportionality: [Laplacian]r∝[Laplacian]t.
In[]:=
simulatePath2[mass_,initialsemimajoraxis_,initialangularmomentum_,maxtime_,mintime_:0,animationrate_:5]:= Modulee,m,L,a,system1,angularMomentum,semiMajorAxis,eccentricity, r,θ,system2,angle,radius,seed,n1,n2,tinystars,smallstars,x, e[t_]:=1(1+(7/8)), a'[t]-435-366a[t]+37, L[0]initialangularmomentum, a[0]initialsemimajoraxis,{L,a},{t,0,maxtime}; angularMomentum[t_]:=L[t]/.system1[[1]]; semiMajorAxis[t_]:=a[t]/.system1[[1]]; eccentricity[t_]:=; system2=NDSolveθ'[t],θ[0]0,θ,{t,0,maxtime}; angle[x_]:=Re[θ[x]/.system2[[1]]]; radius[x_]:=Re; tinystars= 50 Table[{Cos[#1]Sqrt[1-#2^2],Sin[#1]Sqrt[1-#2^2],#2}&[RandomReal[{0,2Pi}],RandomReal[{-1,1}]],{5000}]; smallstars= 45 Table[{Cos[#1]Sqrt[1-#2^2],Sin[#1]Sqrt[1-#2^2],#2}&[RandomReal[{0,2Pi}],RandomReal[{-1,1}]],{500}]; Row[Plot[radius[t],{t,mintime,maxtime},PlotPoints1000,AxesLabel{"time","radius"}, ImageSize{450,400}], Animate[Show[ ParametricPlot3D[(1-m/(1+m))radius[t] Cos[angle[t]], radius[t]Sin[angle[t]],0,(m/(1+m)){-radius[t] Cos[angle[t]],-radius[t] Sin[angle[t]],0}, {t,mintime,maxtime},PlotPoints5000], Graphics3D[{RGBColor[0,0,0],Sphere[(1-m/(1+m)){radius[x] Cos[angle[x]],radius[x] Sin[angle[x]],0}]}], Graphics3D[{RGBColor[0,0,0],Sphere[(m/(1+m)){-radius[x] Cos[angle[x]],-radius[x] Sin[angle[x]],0}]}], ImageSize{450,400}, PlotRangeMax[{(1-m/(1+m)),(m/(1+m))}]{-radius[mintime] ,radius[mintime] }, Max[{(1-m/(1+m)),(m/(1+m))}]{-radius[mintime] ,radius[mintime] }, {-2,2}], {{x,mintime,"time"},mintime,maxtime}, SaveDefinitionsTrue, AnimationRateanimationrate], Animate[Show[ Graphics3D[ RGBColor[1,1,1],PointSize[0.003],Point[tinystars], RGBColor[1,1,1],PointSize[0.006],Point[smallstars]], Graphics3D[{RGBColor[0,0,0],Sphere[(1-m/(1+m)){radius[x] Cos[angle[x]],radius[x] Sin[angle[x]],0}]}], Graphics3D[{RGBColor[0,0,0],Sphere[(m/(1+m)){-radius[x] Cos[angle[x]],-radius[x] Sin[angle[x]],0}]}], BackgroundRGBColor[0.07,0.2,0.40],BoxedFalse,SphericalRegionTrue, ImageSize{450,400},LightingAutomatic,ViewVector{0,0,-20},ViewAngle60°], {{x,mintime,"time"},mintime,maxtime}, SaveDefinitionsTrue, AnimationRateanimationrate]]
1-*a[t]
; m=mass; system1=NDSolveL'[t]-(m+1)*(L[t])^2
2
m
32
7
2
1
2
m
1
2
(1+m)
5
5
1
7
2
a[t]
2
(1-)
2
e[t]
2
e[t]
2
8
m
1
2
a[t]
15
5
2
(1+m)
7
L[t]
(m+1)
2
L[t]
2
m
2
(1+m)
4
L[t]
4
m
2
a[t]
1-*semiMajorAxis[t]
; r[t_]:=(m+1)*(angularMomentum[t])^2
2
m
semiMajorAxis[t]*(1-)
2
(eccentricity[t])
1+eccentricity[t]Cos[θ[t]]
1(1+m)*semiMajorAxis[t]*(1-)
2
eccentricity[t]
2
r[t]
semiMajorAxis[x]*(1-)
2
(eccentricity[x])
1+eccentricity[x]Cos[angle[x]]
For the final animation, a star-field was created which served as the background for the polished visualization (the color of “space” was changed to a dark blue in order to see the orbit of the black holes).
In[]:=
simulatePath2[1.1,20,3.39,1327.49,1200,4]
Out[]=
| ||
 |
| ||
 |
Conclusion
Conclusion
In conclusion, this demonstration was able to successfully visualize a binary black hole merger as discovered by LIGO, and as predicted by Albert Einstein’s General Theory of Relativity in 1916. Through this visualization, thought it is somewhat simple, we can view this incredible phenomenon, and are given a glimpse as to what it may be like if we could see the real event with our own eyes. In the future, a second part of the project would be added to determine the distance of the gravitational wave detector from the black holes, and the time the signal reached the detector .
Cite this as: Macy Maurer Levin, "Visualizing a Binary Black Hole Merger" from the Notebook Archive (2019), https://notebookarchive.org/2019-07-5k0auf5
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