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junk.txt
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\input{Header.tex}
\setlength{\parskip}{0.5em} % Space between paragraphs
\newenvironment{myitemize}
{ \vspace{-\topsep}
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt} }
{ \end{itemize}
\vspace{-\topsep}
}
\setlength{\abovedisplayskip}{-40pt}
\setlength{\belowdisplayskip}{-40pt}
\newcommand{\DEC}{\texttt{DEC}}
\newcommand{\RA}{\texttt{RA}}
\titlespacing*{\section}{0pt}{0pt}{0pt}
\begin{document}
\small
%\begin{minipage}{1.0\textwidth}
\begin{multicols*}{2}
\section*{Astronomical Coordinates}
\textbf{Declination} (\DEC\ or $\delta$) is equivalent to terrestrial
latitude. Points north of the celestial
equator have positive declinations, while those to the south have
negative declinations. The sign is customarily included even if it is
positive. Declination is expressed in degrees [$\,^{\circ}\,$],
arc-minutes [$\,\prime\,$], and arc-seconds [$\,\prime\prime\,$].
1$^{\circ}$ = 60$^{\prime}$ = 3600$^{\prime\prime}$
Example: $\rm{\DEC}$ = +23$^{\circ}$ 52$^{\prime}$ 12.12$^{\prime\prime}$
\begin{myitemize}
\item The celestial equator has a $\rm{\DEC}$ = 0$^{\circ}$
\item The celestial north pole has a $\rm{\DEC}$ = +90$^{\circ}$
\item The celestial south pole has a $\rm{\DEC}$ = -90$^{\circ}$
\end{myitemize}
\textbf{Right Ascension} (\RA\ or $\alpha$) is roughly equivalent to
terrestrial longitude. The units of right ascension are hours, minutes,
seconds [hms].
1 hour in \RA\ = 15$^{\circ}$, 24 hours in \RA\ = 360$^{\circ}$.
Example: $\rm{\RA}$ = 20h 23m 12.12s
The zero-point for right ascension is the position of the Sun on the
first day of spring (Vernal Equinox). \RA\ is measured eastward from the
Vernal equinox. The position of the Sun at the beginning of each season
is given in the table below.
{\centering
\includegraphics[width=0.45\textwidth]{./Images/SunPos.png}
\par
}
\begin{tabular}{lccc}
\toprule
Season & Approx Date & $\rm{\RA}$ & $\rm{\DEC}$ \\
\midrule
Vernal Equinox & Mar 20 & 0h & 0$^{\circ}$ \\
Summer Solstice & Jun 21 & 6h & +23.4$^{\circ}$ \\
Autumnal Equinox & Sep 23 & 12h & 0$^{\circ}$ \\
Winter Solstice & Dec 21 & 18h & -23.4$^{\circ}$ \\
\bottomrule
\end{tabular}
As you can see, the Sun moves about 1h in \RA\ every 2 weeks.
\textbf{Julian Date} (JD) The elapsed time since Jan 1, 4713 BCE at noon. Notice the zero point is noon, not midnight. Measured in seconds. Sept 25, 2019 at noon (JD = 2458752.0)
\section*{Celestial Sphere}
{\centering
\includegraphics[width=0.45\textwidth]{./Images/horizon.pdf}
\par
}
\textbf{Meridian}. The meridian is the great circle running from due
north to south, passing through the celestial pole. The moment an object
is on the meridian, that object will have a right ascension (\RA) equal to the
Local Sidereal Time (LST).
LST = \RA\ of the meridian. Since the zero-point of right ascension
is a fixed point in space, the value of the LST will constantly be
changing (the Earth is rotating), and will be depend on your location on
Earth.
\textbf{Midnight} The LST at any location at local midnight will be the
\RA\ of the Sun + 12 h
\textbf{Zenith}. This is the point on the celestial sphere directly
above the observer's location. It is the point on the meridian with the
highest altitude. An object at the zenith will have the coordinates:
\RA\ = LST, \DEC\ = observer's latitude ($\theta$).
\textbf{Visibility}. The ability of an observer to see an object in the
sky depends on the observer's latitude and longitude ($\theta, \ \phi$),
the time of the observation (LST), and the coordinates of the object
(\RA, \DEC).
Some objects may always be above the observer's horizon. These objects
are called \textbf{circumpolar} objects. Of course you may not be able
to see these objects if the Sun is in the sky. Conversely, some objects
may never rise above the observer's horizon.
\vspace{5mm}
\footnotesize
\begin{tabular}{lcc}
\toprule
Visibility & Northern Observer & Southern Observer \\
& ($\theta>0$) & ($\theta<0$) \\
\midrule
Circumpolar & $\rm{\DEC}\,>\,90^{\circ}-\theta$ & $\rm{\DEC}\,<\,-90^{\circ}-\theta$ \\
Never Visible & $\rm{\DEC}\,<\,-90^{\circ}+\theta$ &
$\rm{\DEC}\,>\,90^{\circ}+\theta$ \\ \\
Best Visibility & \multicolumn{2}{c}{$\rm{\RA}\ =\ $ LST at midnight} \\
\bottomrule
\end{tabular}
\small
\columnbreak
\section*{Orbits}
Kepler's first law says: {\it The orbit of every planet is an ellipse
with the sun at one focus}. The semi-major axis (\textbf{a}) and the
eccentricity (\textbf{ecc}) parametrize the size and shape of the ellipse.
For a closed elliptical orbit (orbits gravitationally bound to the Sun),
$\rm{ecc} = \sqrt{1 - {b^2}/{a^2}}$, where \textbf{a} and \textbf{b} are the semi-major and semi-minor axes (see drawing below).
When a = b, ecc = 0 (a circle), and when a $>>$ b, ecc
approaches 1. When ecc = 1, the orbit is a parabolic orbit (just bound).
When ecc $>$ 1 the orbit is a hyperbolic orbit (unbound).
{\centering
\includegraphics[width=0.40\textwidth]{./Images/Orbit_Sun.png}
\par
}
The point in the object's orbit where it is closest to the Sun is called
the \textbf{perihelion} (periapsis). The farthest point in the orbit is
called the \textbf{aphelion} (apoapsis). Inspecting the diagram at the
right, you can see:
perihelion distance $= \rm{a}\ (1-\rm{ecc})$\\[5pt]
aphelion distance $= \rm{a}\ (1+\rm{ecc})$
\section*{Star Stuff}
\textbf{Brightness} The brightness of a star is a measure of the flux (F) received from the star. The flux depends on the star's intrinsic luminosity (L) and distance (d).
\[
\textrm{F} = \frac{L}{4\pi d^2}
\]
\textbf{Apparent magnitude} (m) is a measure of the relative brightness of a star. The magnitude scale is an inverse logarithmic relation, where a difference of 1.0 in magnitude corresponds to a brightness ratio 2.512. The brighter an object appears, the \textbf{lower} its magnitude.
\[
m_{1}-m_{2}=-2.5\log _{{10}}\left({\frac {F_{1}}{F_{2}}}\right)
\]
\textbf{Absolute magnitude} (M) is defined to be equal to the apparent magnitude of a star if it were viewed from a distance of exactly 10.0 parsecs (32.6 light-years). The Sun has an absolute magnitude of M$_{Sun}$ = +4.83.
\textbf{Distance Modulus} ($\mu$) is a relation between a star's apparent magnitude, absolute magnitude, and distance.
\[
{\displaystyle \mu =m-M=5\log _{10}(d)-5=5\log _{10}\left({\frac {d}{10\,\mathrm {pc} }}\right)}
\]
\textbf{Color Index} A simple numerical expression that determines the color of an object. The smaller the color index, the more blue (or hotter) the object is, the larger the color index, the more red (or cooler) the object is.
To measure the index, one observes the magnitude of an object successively through two different filters, such as B and V. The difference in magnitudes found with these filters is called the B-V color index.
The table below give the typical Absolute magnitude (V filter: M$_V$), and the color index (various filters) for stars on the main sequence.
%\footnotesize
\begin{center}
\begin{tabular}{lrrrrr}
\toprule
Class & M$_V$ & U-B & B-V & V-R & V-I \\
\midrule
O5V & -5.2 & -1.19 & -0.32 & -0.14 & -0.32 \\
O8V & -4.3 & -1.14 & -0.32 & -0.14 & -0.32 \\
B0V & -3.7 & -1.07 & -0.30 & -0.13 & -0.30 \\
B3V & -1.4 & -0.75 & -0.18 & -0.08 & -0.20 \\
B6V & -1.0 & -0.50 & -0.14 & -0.06 & -0.13 \\
B8V & -0.25 & -0.30 & -0.11 & -0.04 & -0.09 \\
A0V & 0.8 & 0.00 & 0.00 & 0.00 & 0.00 \\
A5V & 1.8 & 0.08 & 0.19 & 0.13 & 0.27 \\
F0V & 2.4 & 0.06 & 0.32 & 0.16 & 0.33 \\
F5V & 3.3 & -0.03 & 0.41 & 0.27 & 0.53 \\
G0V & 4.2 & 0.05 & 0.59 & 0.33 & 0.66 \\
Sun & 4.83 & 0.14 & 0.65 & 0.36 & 0.72 \\
G5V & 4.93 & 0.13 & 0.69 & 0.37 & 0.73 \\
K0V & 5.9 & 0.46 & 0.84 & 0.48 & 0.88 \\
K5V & 7.5 & 0.91 & 1.08 & 0.66 & 1.33 \\
K7V & 8.3 & & 1.32 & 0.83 & 1.60 \\
M0V & 8.9 & & 1.41 & 0.89 & 1.80 \\
M2V & 11.2 & & 1.50 & 1.00 & 2.2 \\
M4V & 12.7 & & 1.60 & 1.20 & 2.9 \\
M6V & 16.5 & & & 1.90 & 4.1 \\
\bottomrule
\end{tabular}
\end{center}
%\small
\end{multicols*}
%\end{minipage}
\end{document}
\begin{center}
\begin{tabular}{lrrrrr}
\toprule
Class & B-V & U-B & V-R & R-I & Teff (K) \\
\midrule
O5V & -0.33 & -1.19 & -0.15 & -0.32 & 42,000 \\
B0V & -0.30 & -1.08 & -0.13 & -0.29 & 30,000 \\
A0V & -0.02 & -0.02 & 0.02 & -0.02 & 9,790 \\
F0V & 0.30 & 0.03 & 0.30 & 0.17 & 7,300 \\
G0V & 0.58 & 0.06 & 0.50 & 0.31 & 5,940 \\
K0V & 0.81 & 0.45 & 0.64 & 0.42 & 5,150 \\
M0V & 1.40 & 1.22 & 1.28 & 0.91 & 3,840 \\
\bottomrule
\end{tabular}
\end{center}
{\centering
\includegraphics[width=0.70\textwidth]{./Images/StarColor.png}
\par
}
% Just a comment at the end.