1. Visualizing V1309 Scorpii: A Photorealistic
Approach to Simulating a Binary Merger
Kyle Donnelly
January 4, 2017
A multimillenial pastime, astronomy represents the endeavor of hu-
mans to understand the surrounding world at its grandest scale. Astronomy
differs from other branches of natural science in that the objects within its
purview (stars, planets, etc.) are far too large and distant to experiment with
directly. Indeed, up until the invention of the computer, experimentation in
astronomy relied on circumstance; i.e., astronomers could make observations of
and predictions about the natural world, but had no control of the pre-existing
conditions of their predictions. However, following the advent of computational
physics, scientists were able to test astronomical theories, by way of computer
simulation, in a more direct way.
A recent triumph within astrophysics is the description of the con-
ditions leading to a sudden luminous outburst by the star V1309 Scorpii. A
formerly dim and static star, it was discovered in 2008 (Nakano 2008) only as
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2. a result of serendipity, its position being in the line of sight of astronomers
observing the galactic center. After its violent outburst, in which it grew to
thousands of times its original brightness. Tylenda et al. were delighted to find
that they had been unknowingly gathering data on this star since 2002, and the
task remained to explain its behavior. Up until its eruption, the star demon-
strated periodic variability in brightness, and after its eruption, it remained
dim and unchanging in brightness. The sequence of variability, outburst, and
static brightness was baffling to the scientific community, until Tylenda et al.
purported that this event, and events similar to it (such as the outburst of V838
Monocerotis), were the result of the union of two stars (Tylenda 2011). Ac-
cording to this assertion, the stars orbited their common center of mass (two
stars that orbit in this way are jointly referred to as a binary star system), until
they united, releasing energy and matter, and leaving behind a single star af-
terwards. What remains is for computational astrophysicists to quantify initial
conditions about the system (such as initial separation, masses, temperature,
separation etc.) using computer simulation. Due to the plethora of available
data surrounding V1309 Sco and its merger, the star system remains a prime
subject for the study of binary star systems and, especially, stellar mergers.
The only parameter with which astrophysicists can verify their simu-
lations of V1309 Sco is the light curve (a plot showing brightness versus time,
as shown below in Figure 1) from the time before and after the merger event.
Most work on V1309 has been in attempting to mimic the initial periodicity in
brightness, as well as a peak in brightness corresponding to the merger event,
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3. Figure 1: The light curve of V1309 Sco; magnitude of the star is plotted against
the time of observation in (Julian) days, with years on the top of the figure.
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4. but conclusive results have not yet been found (Nandez et al. 2014, Zhu et al.
2016). However, Wenskovitch et al. have developed a tool more suited to the
accurate visual recreation of stellar events, by allowing the user to superpose
images at different wavelengths for a more visually accurate visualization. It
still remains, in the pursuit of photorealism, to determine the most suitable
wavelength filters (based on the spectral sensitivity of cone cells of the human
eye).
The human eye contains two types of photoreceptors (cells that detect
light): rods, which allow vision at low light, and cones, which allow for color
vision. The color vision cells are the short cone, middle cone, and long cone.
Since there are only three cells, one can describe any perceivable color by the
degree of stimulation of each cell. In 1931, the International Commission on
Illumunatio (CIE) produced functions that represent the responsiveness of cones
to light as a function of wavelength. These three functions are collectively
called the CIE 1931 Colour Matching functions, ¯x(λ), ¯y(λ), and ¯z(λ), which are
depicted in Figure 2.
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5. Figure 2: Plots of the CIE color matching functions versus wavelength. The ¯y(λ)
curve represents brightness at a given wavelength, the ¯z(λ) curve contributes
some of the sensitivity of short and medium cones, and the ¯x(λ) contributes
sensitivity of short, medium, and long cones.
These can be multipled by a spectral power distribution (the power of
emitted light versus wavelength) and then integrated to give three the tristim-
ulus values X, Y , and Z, which can be conceptualized as the amount of each
primary color in a ray of light, and can be mapped to a color. Thus, each color
can be described by a unique set of tristimulus values.
X = ¯x(λ)Lλdλ
Y = ¯y(λ)Lλdλ
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6. Z = ¯z(λ)Lλdλ
The tristimulus values are used to define a pair of normalized coor-
dinates called the chromaticity coordinates (x, y), which uniquely describe any
perceivable color. They are obtained by the following relations:
x =
X
X + Y + Z
y =
Y
X + Y + Z
Mapping every perceivable color to a chromaticity coordinate, one can constuct
a plot of the color perceived at each coordinate, called a color space, shown
below, in Figure 3 (Wyszecki 1967).
Many phenomena in cosmology and astrophysics involve fluid dynam-
ics in their understanding. When attempting to create a computer simulation
of the gas involved in stellar events, the most widely used method is that of
Smoothed Particle Hydrodynamics (SPH). SPH is Lagrangian method, which
means it partitions a fluid into a finite number of parcels, which interact with
pressure forces with other parcels within a radius away (which is calculated for
each particle based on number of nearby parcels), and gravitationally with each
other parcel. The acceleration of each particle is given by
a = −
P
ρ
+ g.
Each parcel also has position, mass, velocity, and specific internal energy, which
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7. Figure 3: Depicted is the CIE 1931 color space, representing the gamut of
human color perception and how color relates to the corresponding chromaticity
coordinates. The values on the edge of the gamut (ranging 460-620) indicate
the perceived color of monochromatic light at various wavelengths.
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8. are modified as the simulation progresses after each timestep (the finite length
of time over which the simulation is allowed to evolve before recalculation) ac-
cording to the laws of thermodynamics and Newton’s laws. Because parcels only
exert pressure forces on other nearby parcels within the simulation, instead of
all other matter, SPH is acclaimed for its efficiency, while still conserving energy
and momentum exceptionally well ; this is largely due to the fact that pressure
interactions are negligible over large distances, so that the the summation of
all local interactions indicate the net behavior of the entire system, even in
physically complicated scenarios. Considering the strengths of the SPH, it is
unsurprising that the computer code used in our lab, Starsmasher, makes use
of it. (Gaburov et al. 2010, Rosswog 2009).
There are three types of simulations one can run when using Stars-
masher. The first is called a relaxation run. The code involves fixing a number
of SPH particles initially (which themselves do not represent particles in physi-
cal sense, but are parcels of fluid, the fundamental units of an SPH simulation)
in a hexagonal close packed lattice and allowing the internal pressure and grav-
itational forces to cause them oscillate until each reaches equilibrium, at which
point the simulation ends. While the simulation proceeds, the total energy of
the system is recorded. By the end of a successful run, the total energy should
have very little variation over time. The second type of simulation is called
a scanning run, in which two stars (that were produced in relaxation runs)
are moved a small distance closer together after each timestep, until the user-
specified final separation is reached, at which the simulation ends. Energy from
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9. this simulation will not remain constant because work outside the star system
is being done to force the stars close together. The final type of simulation is a
dynamical run, in which, after the user is prompted for initial conditions about
the star system, the simulation will proceed to treat the system represented as
if it were physical, conserving energy and momentum, and obeying Newton’s
laws. The simulation ends after a prespecified amount of simulated time has
passed.
A blackbody is a theoretical object that absorbs all light and does not
reflect any. It emits radiation depending on its temperature, with an energy
spectrum Bλ given by
Bλ(λ, T) =
2hc2
λ5
1
ehc/λkBT − 1
where λ represents radiation wavelength, h represents Planck’s constant, c repre-
sents the speed of light in a vacuum, kB represents the Boltzmann constant, and
Bλ has units of power per area per unit solid angle per wavelength (W/m3
/sr
in SI units). This is known as Planck’s law. In terms of frequency ν, it reads:
Bν(ν, T) =
2hν3
c2
1
ehν/kBT − 1
,
in which case the units of Bν would be power per area per frequency per unit
solid angle (W · s/m2
/sr). Planck’s law can approximate the spectrum for
objects made of dense gas, or of an opaque liquid or gas. A blackbody will have
a total flux F (over all wavelengths) given by the Stefan-Boltzmann law:
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10. Figure 4: A typical output file showing the various energies of a star within a
simulation over time as it is relaxed. In descending order, the energies depicted
are specific internal energy, total gravitational potential energy, total kinetic
energy, and total energy. The units used in the Starsmasher code and displayed
in the figure are such that G = Rsol = Msol = 1, where G is Newton’s grav-
itational constant, and Rsol and Msol are the radius and mass of Earth’s sun.
Thus, the unit of time is tu = (
R3
sol
GMsol
)1/2
= 1592s and the unit of energy is
Eu =
GM2
sol
Rsol
= 3.79 · 1041
J.
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11. F = σT4
where σ is referred to as the Stefan-Boltzmann constant. Although it is conve-
nient to represent a star as a blackbody, a star has internal structure which can
scatter radiation propagating from within the star as it leaves. The likelihood
that a ray originating inside a star will be scattered increases with the depth
from which it is propagating. The optical depth is a dimensionless quantity that
gives a measure of how much light will be scattered while traveling through the
star. It can be calculated by solving the differential equation
dτ
dz
= −ρ(z)κ(z)
where ρ(z) is the density at a location z, and κ(z) is the opacity at a location z.
We can then write the intensity of light I, with initial intensity I0, as a function
of the optical depth in the following way:
I(z) = I0e−τ
.
This is called the equation of radiative transfer. We see that the intensity of
light traveling through a gas decreases exponentially with the optical depth.
Large values of τ will indicate an opaque object, while small values will indicate
a transparent object. We also see that e−τ
gives the fraction of light particles
able to travel through a gas from an optical depth of τ. Therefore, we can divide
a column of gas into infinitesimal slabs, each contributing an infinitesimal flux
dF given by
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12. dF = σT4
e−τ
dτ.
Therefore, we can find the total flux by
F = σT4
e−τ
dτ.
Similarly, we can find the flux density of an infinitesimal slab of gas by
dSν = Bν(ν, T)e−τ
dτ
so that
Sν = Bν(ν, T)e−τ
dτ.
There are numerous tools for visualizing the data generated by SPH
simulations but none before FluxE, a program developed by Wenskovitch et al.,
have been made with the express goal of photorealism. The spectral data taken
as input by FluxE to generate visualizations are obtained from a Starsmasher
simulation: The Starsmasher code projects a three-dimensional body of gas into
a grid, and concatenates the flux and flux density of a region of that grid, based
on its local temperature (as if each grid region acted locally as a blackbody),
by solving the following differential equations within each region of the grid:
dF
dz
= σT4
e−τ dτ
dz
dSν
dz
= Bνe−τ dτ
dz
Where F is the flux, Sv is the flux density (flux at a given frequency), Bv is
the Planck function, τ is the optical depth, and σ is the Stefan-Boltzmann con-
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13. stant. For each region of the grid, there are two ways in which the color is
determined. The first involves the definition of a new quantity, called the ef-
fective temperature, Teff , which represents the temperature a blackbody would
be if it had the same flux as is contained within the region. Using Teff , FluxE
uses a database (www.vendian.org/mncharity/dir3/blackbody) which indicates
the color a blackbody assumes at different temperatures, and the region is filled
with that color. The other way to color a region is to to choose up to three
different wavelength filters (so that only the flux of one wavelength of light is
displayed), which can then be superposed to render a composite image, as in
Figure 5 (Wenskovitch et al. 2016).
Figure 5: A diagram showing the use of FluxE to visualize a simulation of V1309
Sco’s merger. Three wavelength filters (colored red, blue, and green) correspond
to the respective intensities of prespecified wavelenghts of light. The system is
shown at various points in its evolution.
In the pursuit of photorealism, to determine the color and brightness
of emitted light in a region, one needs to consider not only the flux therein, but
also how sensitive the eye would to be in detecting the light emitted. Hence,
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14. we can use the flux density at each grid cell, which is the input for FluxE, and
integrate it against the color matching functions to determine the appropriate
color of the grid cell, which can then be rendered within FluxE. Considering
both the entire spectrum of a star as well as the eye’s sensitivity to wavelength
of light should give a more accurate visualization than superposing the images
from three ’representative’ wavelength filters; however, this method will be more
computationally costly.
Attached is the program I have written, which receives a file called
’tri.txt’ containing the color matching functions obtained from http://cvrl.ioo.ucl.ac.uk/cmfs.htm
and stores them in arrays; and a file called ’input.txt’ containing the spectrum
for which one wishes to determine the color, computes integrals the to compute
the tristimulus values, which it converts to chromaticity coordinates, and then
to values of R, G, and B.
Although my program returns values of R, G, and B, comparison
to values obtained by Charity (www.vendian.org/mncharity/dir3/blackbody)
shows some disagreement. This is because Charity transforms RGB values using
a method called ’gamma correction.’ More research is needed on this method,
and a program attempting to utilize it will be written. Next semester, I will
need to write a computer routine accomplishing the same task as my program to
determine RGB values from a spectrum obtained from a Starsmasher simulation,
and use these as input into FluxE to produce more photorealistic visualizations.
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15. References
[1] Nakano, S et al. (2008) V1309 Scorpii = Nova Scorpii 2008.
[2] Tylenda et al. (2011) V1309 Scorpii: merger of a contact binary.
[3] Nandez et al. (2014) V1309 Scorpii-Understanding a Merger.
[4] Zhu et al. (2016) A low-mass-ratio and deep contact binary as the progenitor
of the merger V1309 Sco.
[5] Wyszecki, Gunter Color Science: concepts and methods, quantitative data
and formulas. New York. 1967
[6] Gaburov et al. (2010) On the onset of runaway stellar collisions in dense
star clusters - II. Hydrodynamics of three-body interactions.
[7] Rosswog et al. (2009) Astrophysical smooth particle hydrodynamics.
[8] Charity, M (2016) What is a blackbody?.
www.vendian.org/mncharity/dir3/blackbody
[9] Wenskovitch et al. (2016) FluxE: A Flux Explorer for Visualizing Astrophys-
ical Simulations.
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