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\begin{document}
\title{Hertzsprung-Russell Diagrams: Nuclear Fusion Reactions in the Main
Sequence~}
\author{Armandt Erasmus}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\sloppy
\section*{Introduction}
{\label{348809}}
During the nineteenth-century, the Harvard College Observatory performed
many photographic spectroscopic surveys of stars. This produced spectral
classifications for roughly 225,000 stars, which created the Henry
Draper Catalogue.
Ejnar Hertzsprung realised that narrow lined stars had smaller proper
motions compared to other stars of the same spectral classification. He
was then able to estimate the stars' absolute
magnitudes\textsuperscript{\hyperref[csl:1]{(``{1}'', n.d.)}} by computing their
secular parallaxes as a result of the difference of proper motions.
\section*{Definitions}
{\label{173892}}
Before we start constructing and analysing a Hertzsprung-Russell
Diagram, we should familiarise ourselves with some of the words used in
this paper.
Firstly, Absolute Magnitude (\(M\)) is a measure of the
luminosity of a celestial object, on an inverse logarithmic astronomical
magnitude scale. It is further defined to be equal to the apparent
magnitude that the object would have if it were viewed from a distance
of exactly 10 parsecs without extinction of its light.
Apparent Magnitude (\(m\)) is a measure of the brightness
of a celestial object observed from Earth. This value is dependent on
factors such as the extinction of the body's light, its intrinsic
luminosity and its distance from Earth. We state a relationship, the
brighter an object, the lower its magnitude value.
Luminosity (\(L\)\textsubscript{\(\circ\)} or
\(L\)\textsubscript{*}) refers to the absolute measure of
radiated electromagnetic power emitted by a light-emitting object. We
are interested in both the solar luminosity, denoted by
\(L\)\textsubscript{\(\circ\)} and the stellar
luminosity, denoted by \(L\)\textsubscript{*} of a
particular star.
Lastly, we define the Hertzsprung-Russell Diagram as a scatter plot of
stars which shows the relationship between the stars' absolute
magnitudes or luminosities versus their stellar classifications or
effective temperatures.
\section*{Absolute Magnitude}
{\label{454892}}
We know that absolute magnitude is measured by a body's luminosity,
which provides us with the relationship between Absolute Magnitude and
Luminosity, which states that the more luminous an object is, the
smaller the numerical value of its absolute magnitude. In terms of
variables, we can express this relationship by the following:
A difference of~\emph{n} magnitudes ( in absolute magnitude )
corresponds to a luminosity ratio of \(100^{\frac{n}{5}}\).~
In a general sense, subscripts are used alongside~\(M\) to
represent the filter band used for the specific measurement,
e.g.~\(M\)\textsubscript{\emph{V}}~for measurements in the
V-Band. We generalise this over all wavelengths with an object's
bolometric magnitude. By applying a bolometric
correction\textsuperscript{\hyperref[csl:2]{(``{2}'', n.d.)}}, we can convert absolute
magnitudes in specific filter bands to its absolute bolometric
magnitude.
\[M_{bol}=M_v+BC\]
Where BC is the Bolometric Correction, needed to factor in specific
types of radiation by celestial bodies.
\section*{Apparent Magnitude}
{\label{726214}}
A numerical scale by Hipparchus describes the brightness of stars that
appear in the night sky, where~\(m\) = 1 is assigned to
the brightest stars and~\(m\) =
6\textsuperscript{\hyperref[csl:3]{(``{3}'', n.d.)}} assigned to the dimmest stars.
The equation,
\[100^{\frac{mM}{5}}=\frac{F}{F_{10}}=\left(\frac{d}{10pc}\right)^2\]
relates objects within the neighbourhood of the Sun, where their
brightness differs by a factor of 100 for \(m\) and
\(M\) from any distance \(d\).
The following equation is derived, given that~\(d\) is
measured in parsecs,
\[M=m-5\log_{10}\left(d_{pc}\right)+5\]~
Which when simplified produces,
\[M=m-5\left(\log_{10}d_{pc}-1\right)\]
This equation can also be written in two other forms,
In terms of stellar parallax.
(1) \[M=m+5\left(\log_{10}p+1\right)\]
Where a distance modulus is known:
(2) \[M=m-\mu\]
For this paper, we will only be concerned with equation (1).
\par\null
\section*{Gathering Data}
{\label{385646}}
The following table was generated by using data available
from~\href{http://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=I/239/hip_main}{VizieR}.
The Vmag column represents the H5 V Johnson magnitudes, Plx the
Trigonometric Parallax, B-V the Johnson B-V Colours and finally SpType
to represent all the Spectral Types.\selectlanguage{english}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
HIP Identification & Magnitude in Johnson V H5 (Vmag) & Trigonometric Parallax H11 (Plx) & Johnson B-V Colour H37 (B-V) & Spectral Type H76 \\ \hline
1 & 9.10 & 3.54 & 0.482 & F5 \\ \hline
2 & 9.27 & 21.90 & 0.999 & K3V \\ \hline
3 & 6.61 & 2.81 & -0.019 & B9 \\ \hline
4 & 8.06 & 7.75 & 0.370 & F0V \\ \hline
5 & 8.55 & 2.87 & 0.902 & G8III \\ \hline
... & ... & ... & ... & ... \\ \hline
10003 & 8.45 & -0.93 & 1.404 & K5 \\ \hline
10004 & 7.84 & 4.26 & 1.14 & K1IIICN \\ \hline
10005 & 9.38 & 3.61 & 0.507 & G0 \\ \hline
10006 & 7.64 & 4.75 & 0.075 & A2 \\ \hline
10007 & 9.00 & 10.61 & 0.566 & G1V \\ \hline
\end{tabular}
\end{table}
We start by creating an additional column, M\textsubscript{v}, to
represent the absolute magnitudes in the V-Filter.
This is done by using equation (1) in such form,
\[M_V=V_{mag}+5\left(\log_{10}\ \frac{plx}{100}\right)\]
This produces:\selectlanguage{english}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
HIP Identification & Magnitude in Johnson V H5 (Vmag) & Trigonometric Parallax H11 (Plx) & Johnson B-V Colour H37 (B-V) & Spectral Type H76 & Absolute Magnitude \\ \hline
1 & 9.10 & 3.54 & 0.482 & F5 & 1.845016 \\ \hline
2 & 9.27 & 21.90 & 0.999 & K3V & 5.972221 \\ \hline
3 & 6.61 & 2.81 & -0.019 & B9 & -1.146468 \\ \hline
4 & 8.06 & 7.75 & 0.370 & F0V & 2.506509 \\ \hline
5 & 8.55 & 2.87 & 0.902 & G8III & 0.839409 \\ \hline
... & ... & ... & ... & ... & ... \\ \hline
\end{tabular}
\end{table}
By plotting an Absolute Magnitude versus Spectral Class diagram, we
obtain a simplified Hertzsprung-Russel Diagram as shown below.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/Data3/PreDataGraph}
\caption{{Simplified Hertzsprung-Russell Diagram.
{\label{507295}}%
}}
\end{center}
\end{figure}
Although the plotted values do not fill much space, it starts to
represent the basic pattern you would expect from a~Hertzsprung-Russell
Diagram, especially the formation of the Main-Sequence.
Now, by plotting Absolute Magnitude values against Colour Index.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/DataGraph/DataGraph}
\caption{{Improved Hertzsprung-Russell Diagram constructed with data from 9000+
stars/
{\label{734393}}%
}}
\end{center}
\end{figure}
Finally,~ we can create a coloured diagram to represent different star
types to explain the Hertzsprung-Russell
Diagram\textsuperscript{\hyperref[csl:4]{(``{4}'', n.d.)}}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/PostDataGraph/PostDataGraph}
\caption{{Generated Hertzsprung-Russell Diagram
{\label{357132}}%
}}
\end{center}
\end{figure}
\section*{Interpreting the Hertzsprung-Russell
Diagram}
{\label{236217}}
The purple area shows the main sequence, where stars spend most of their
lives. Stars located here undergo nuclear fusion, where they fuse
Hydrogen into their cores while also remaining stable. During nuclear
fusion, Hydrogen is converted into Helium.
This reaction can very easily be described by the Proton-Proton Chain
Reaction where two protons fuse together to produce Deuterium. Deuterium
is a stable isotope of Hydrogen and is given by the following chemical
equation,
\[p\ +\ p\ \to\ _1^2D\ +\ e^++v_e+1.442\ MeV\]
Upon completion, the Deuterium can fuse with another proton to produce
the light isotope of Hydrogen. This reaction is given by,
\[_1^2D\ +\ _1^1H\ \to_2^3He+\gamma+5.49\ MeV\]
To the right of the diagram, we see the Red Giants whose cores' Hydrogen
have been exhausted. The Red Giants have lower temperatures than the
stars located in the main sequence but have higher stellar luminosity
values.
The final stage presented by a Hertzsprung-Russell Diagram is the
formation of White Dwarfs, but since our data did not include White
Dwarfs we cannot explain them.
\section*{The Proton-Proton Cycle}
{\label{412536}}
We already discussed the basic chemical equation by which two protons
produce Deuterium. Now we will study this process in greater detail.
Our two protons can be seen as two~\(_1^1H\) atoms, and we can
rewrite the chemical equation as,
\[_1^1H+\ _1^1H\to\ _1^2D\ +e^++v_e\]
Where~\(e^+\) is the Positron and~\(v_e\) the
Neutrino.
Due to the collision between electrons and positrons, annihilation
occurs whereby two other particles are produced.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.42\columnwidth]{figures/Mutual-Annihilation-of-a-Positron-Electron-pair-svg/Mutual-Annihilation-of-a-Positron-Electron-pair}
\caption{{Annihilation of an electron-positron pair given by a Feynman
Diagram\textsuperscript{\protect\hyperref[csl:5]{(``{5}'', n.d.)}}.
{\label{337075}}%
}}
\end{center}
\end{figure}
\section*{Electron-Positron
Annihilation}
{\label{412536}}
Positrons produced from equation (10) do not last very long as they
collide with electrons. The electron annihilates the positron to produce
two photons or more specifically two Gamma Rays.
This reaction is given by,
\[e^++e^-\ \to2\gamma\]
Energy is also produced in Megaelectronvolts, and amounts to 0.511 MeV.
This energy contributes to the total value of released energy from
equation (10).
The total energy released sums to 1.442 MeV such that Q=1.442 MeV
\section*{Deuterium-Proton Fusion}
{\label{193862}}
The second part of the products also undergo a separate reaction whereby
the Deuterium fuses with another proton in order to
produce~\(_2^3He\) and is given by,
\[_1^2H\ +_1^1H\to_2^3He+\gamma\]
We also notice a change in energy where the total energy released
increases due to the loss in mass which obeys the mass-energy
equivalence\textsuperscript{\hyperref[csl:6]{(``{6}'', n.d.)}} equation given
by~\(\Delta{}E=\Delta{}mc^2\).
The binding energy equation follows as,
\[2.22452\ MeV\ +0.0000136\ MeV\ \to7.7181\ MeV\]
We see that mass is converted into energy.
\section*{The Proton-Proton 1 Branch}
{\label{678691}}
Our final fusion reaction includes two~\(_2^3He\) atoms fusing
together to produce~\(_2^4He\). The chemical equation is given
by,
\[_2^3He\ +\ _2^3He\to_2^4He\ +_1^1H+\ _1^1H+\gamma\]
The total energy produced by the Proton-Proton Chain equates to 26.7
MeV. Some energy is also lost to neutrinos.
\section*{Conclusion}
{\label{412536}}
The Hertzsprung-Russell Diagram can be used as an accurate method to
approximate the distance between different Galaxies and Star Clusters
from the Earth. We are also further able to explain the nuclear physics
behind the stars found in the main sequence and find that conservation
of mass is at play during fusion.
We are also able to get a better understanding of the Proton-Proton
Chain.
\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\phantomsection
\label{csl:1}\textit{(1) Astronomische Nachrichten}. \url{https://zenodo.org/record/1424859}
\phantomsection
\label{csl:2}\textit{(2) Transformations from Theoretical Hertzsprung-Russell Diagrams to Color-Magnitude Diagrams: Effective Temperatures, B-V Colors, and Bolometric Corrections}. \url{https://en.wikipedia.org/wiki/The_Astrophysical_Journal}
\phantomsection
\label{csl:3}\textit{(3) An Introduction to Modern Astrophysics}. \url{https://archive.org/details/introductiontomo00bwca_613}
\phantomsection
\label{csl:4}\textit{(4) The Hertzsprungâ€“Russell Diagram}. \url{https://en.wikipedia.org/wiki/Hertzsprung\%E2\%80\%93Russell_diagram}
\phantomsection
\label{csl:5}\textit{(5) Feynman Diagrams}. \url{https://en.wikipedia.org/wiki/Feynman_diagram}
\phantomsection
\label{csl:6}\textit{(6) Mass-Energy Equivalence}. \url{https://en.wikipedia.org/wiki/Mass\%E2\%80\%93energy_equivalence}
\end{document}