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\begin{Large}
\section{\bf Characterizing Exoplanets}
\end{Large}

\subsection{Introduction}	

Exoplanets are a hot topic in astronomy right now. As of January, 2015, there 
were over 1500 known exoplanets with more than 3000 candidates 
waiting to be confirmed. These exoplanets and exoplanet systems are of 
great interest to astronomers as they provide information on planet formation 
and evolution, as well as the discovery of a variety of types of planets  
not found in our solar system. A small subset of these planetary systems are 
of interest for another reason: They may support life. In this lab you will 
analyze observations of exoplanets to fully
characterize their nature. At the end, you will then compare your results
with simulated images of these exoplanets to see how well you performed.
Note that the capabilities required to intensely study exoplanets have not 
yet been built and launched into space. But we know enough about optics that 
we can envision a day when advanced space telescopes, like those needed for 
the conclusion of today's lab, will be in Earth orbit and will directly image 
these objects, as well as obtain spectra to search for the chemical signatures 
of life.
	
\subsection{Types of Exoplanets}
	
As you have learned in class this semester, our solar system has two main
types of planets: Terrestrial (rocky) and Jovian (gaseous). Because these
were the only planets we knew about, it was hard to envision what other kinds
of planets might exist. Thus, when the first exoplanet was discovered,
it was a shock for astronomers to find out that this object was a gas
giant like Jupiter, but had an orbit that was even smaller than that of Mercury!
This lead to a new kind of planet called ``Hot Jupiters''. In the two decades
since the discovery of that first exoplanet, several other new types of
planets have been recognized. Currently there are six major classes that we
list below. We expect that other types of planets will be discovered as our 
observational techniques improve.

\subsubsection{Gas Giants}
	
Gas giants are planets similar to Jupiter, Saturn, Uranus, and Neptune. They 
are mostly composed of hydrogen and helium with possible rocky or icy cores. 
Gas giants have masses greater than 10 Earth masses. Roughly 25 percent of 
all discovered exoplanets are gas giants.
	
\subsubsection{Hot Jupiters}
	
Hot Jupiters are gas giants that either formed very close to their host star 
or formed farther out and ``migrated'' inward. If there are multiple planets
orbiting a star, they can interact through their gravity. This means that 
planets can exchange energy, causing their orbits to expand or to shrink.
Astronomers call this process migration, and we believe it happened early
in the history of our own solar system. Hot Jupiters are found within 0.05-0.5 
AU of their host star (remember that the Earth is at 1 AU!). As such, they are 
extremely hot (with temperatures as high as 2400 K), and are the most common 
type of exoplanet found; about 50 percent of all discovered exoplanets are 
Hot Jupiters. This is due to the fact that the easiest exoplanets to detect 
are those that are close to their host star and very large. Hot Jupiters are 
both.
	
\subsubsection{Water Worlds}
	
Water worlds are exoplanets that are completely covered in water. Simulations 
suggest that these planets actually formed from debris rich in ice further 
from their host star. As they migrated inward, the water melted and covered the 
planet in a giant ocean.
	
\subsubsection{Exo-Earths}
	
Exo-Earths are planets just like the Earth. They have a similar mass, radius, 
and temperature to the Earth, orbiting within the ``habitable zone'' of their 
host stars. Only a very small number of Exo-Earth candidates have been 
discovered as they are the hardest type of planet to discover.

\subsubsection{Super-Earths}
	
Super-Earths are potentially rocky planets that have a mass greater than the 
Earth, but no more than 10 times the mass of the Earth. ``Super'' only refers 
to the mass of the planet and has nothing to do with anything else. Therefore, 
some Super Earths may actually be gas planets similar to (slightly)
smaller versions of Uranus or Neptune.
	
\subsubsection{Chthonian Planets}
	
``Chthonian'' is from the Greek meaning ``of the Earth.'' Chthonian Planets 
are exoplanets that used to be gas giants but migrated so close 
to their host star that their atmosphere was stripped away leaving only a 
rocky core. Due to their similarities, some Super Earths may actually be 
Chthonian Planets.
	
\subsection{Detection Methods}
There are several methods used to detect exoplanets. The most useful ones are 
listed below.
	
\subsubsection{Transit Method/Light Curves}

The transit method attempts to detect the ``eclipse'' of a star by a
planet that is orbiting it. Because planets are tiny compared to their
host stars, these eclipses are very small, requiring extremely precise 
measurements. This is best done from space, where observations can be
made continuously, as there
is no night or day, or clouds to get in the way. This is the detection 
method used by the {\it Kepler} Space Telescope. {\it Kepler} stared at a 
particular patch of sky and observed over a hundred thousand stars continuously 
for more than four years. It measured the amount light coming from each star. 
It did this over and over, making a new measurement every 30 minutes. 
Why? If we were looking back at the Sun and wanted to detect the 
Earth, we would only see one transit per year! Thus, you have to continuously
stare at the star to insure you do not miss this event (as you need at least
three of these events to determine that the exoplanet is real, and to
measure its orbital period). The end result is something called a ``light
curve'', a graph of the brightness of a star over time. The entire process
is diagrammed in Figure \ref{exotransit}. We will be exclusively using this 
method in lab today.\\
	
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{Exoplanets/exotransit.jpg}
\caption{The diagram of an exoplanet transit. The planet, small, dark 
circle, crosses in front of the star as seen from Earth. In the process, 
it blocks out some light. The light curve, shown on the bottom, is a plot of 
brightness versus time, and shows that the star brightness is steady until the 
exoplanet starts to cover up some of the visible surface of the star. As it 
does so, the star dims. It eventually returns back to its normal brightness 
only to await the next transit.}
\label{exotransit}
\end{figure}
	
In Figure \ref{exotransit}, there is a dip in the light curve, signifying that an object 
passed between the star and our line of sight. If, however, {\it Kepler} 
continues 
to observe that star and sees the same sized dip in the light curve on a 
periodic basis, then it has probably detected an exoplanet (we say ``probably''
because a few other conditions must be met for it to be a confirmed exoplanet).
The amount of star light removed by the planet is very small, as all planets
are much, much smaller than their host stars (for example, the radius of
Jupiter is 11 times that of the Earth, but it is only 10\% the radius of the
Sun, {\it or 1\% of the area = how much the light dims}). Therefore, it is 
much easier to detect planets that are larger because they block more of the 
light from the star. It is also easier to detect planets that are close to 
their host star because they orbit quickly so {\it Kepler} could observe 
several dips in the light curve each year. \\
	
\subsubsection{Direct Detection}
	
Direct detection is exactly what it sounds like. This is the method of 
imaging (taking a picture) of the planets around another star. But we cannot 
simply point a 
telescope at a star and take a picture because the star is anywhere from 100 
million ($10^8$) to 100 billion ($10^{11}$) times brighter than its
exoplanets. In order to combat the overwhelming brightness of a star, 
astronomers use what is called a ``coronagraph'' to block the light from the 
star in order to see the planets around it. You may have already seen images 
made with a coronagraph to see the ``corona'' of the Sun in the Sun lab. \\
	
\begin{figure}[h!]
\centering
\includegraphics[scale=0.25]{Exoplanets/Fomalhaut_B.jpg}
\caption{A coronagraphic image of an exoplanet orbiting the star Fomalhaut
(inside the box, with the arrow labeled ``2012''). 
This image was obtained with the Hubble
Space Telescope, and the star's light has been blocked-out using a small metal 
disk. Fomalhaut is also surrounded by a dusty disk of material---the broad 
band of light that makes a complete circle around the star. This band of dusty 
material is about the same size as the Kuiper belt in our solar system. The 
planet, ``Fomalhaut B'', is estimated to take 1,700 years to orbit once around 
the star. Thus, using Kepler's third law (P$^{\rm 2}$ $\propto$ a$^{\rm 3}$),
it is roughly about 140 AU from Fomalhaut (remember that Pluto orbits at 
39.5 AU from the Sun).}
\label{fomalhaut}
  \end{figure}
		
So if astronomers can block the light from the Sun to see its corona, they 
should be able to block the light from distant stars to see the exoplanets 
right? While this is true, directly seeing exoplanets is difficult. There
are two problems: the exoplanet only shines by reflected light, and it 
is located very, very close to its host star. Thus, it takes highly 
specialized techniques to directly image exoplanets. However, for some of the 
closest stars this can be done. An example of direct exoplanet detection is 
shown in Figure \ref{fomalhaut}. A new generation of space-based telescopes 
that will allow us to do this for many more stars is planned. Eventually, we
should be able to take both spectra (to determine their composition) and
direct images of the planets themselves. We will pretend that we can obtain
good images of exoplanets later in lab today.
	
\subsubsection{Radial Velocity (Stellar Wobble)}
	
The radial velocity or ``stellar wobble" method involves measuring the Doppler 
shift of the light from a particular star and seeing if the lines in its 
spectrum oscillate periodically between a red and blue shift. As a planet 
orbits its star, the planet pulls on the star gravitationally just as the star 
pulls on the planet. Thus, as the planet goes around and around, it slightly 
tugs on the star and makes it wobble, causing a back and forth shift in its 
radial velocity, the motion we see towards and away from us. Therefore, if 
astronomers see a star wobbling back and forth on a repeating, periodic 
timescale, then the star has at least one planet orbiting around it. The size 
of the wobble allows astronomers to calculate the mass of the exoplanet.\\
	
\subsection{Characterizing Exoplanets from Transit Light Curves}

Quite a bit of information about an exoplanet can be gleaned from its
transit light curve. Figure \ref{transit} shows how a little bit of math
(from Kepler's laws), and a few measurements, can tell us much about a
transiting exoplanet.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{Exoplanets/Transit.jpg}
\caption{An exoplanet transit light curve (bottom) can provide
a useful amount of information. The most important
attribute is the radius of the exoplanet. But if you know the mass and
radius of the exoplanet host star, you can determine other details about the
exoplanet's orbit. As the figure suggests, by observing multiple transits
of an exoplanet, you can actually determine whether it has a moon! This
is because the exoplanet and its moon orbit around the center of mass
of the system (``barycenter''), and thus the planet appears to wobble back 
and forth relative to the host star.}
\label{transit}
\end{figure}

The equations shown in Figure \ref{transit} are complicated by the fact
that exoplanets do not orbit their host stars in perfect circles, and
that the transit is never exactly centered. Today we are going to only
study planets that have circular orbits, and whose orbital plane is edge-on.
Thus, all of the terms with ``cos$i$'' (``$i$'' is the inclination of
the orbit to our sight line, and $i$ = 0$^{\circ}$ for edge on), cos$\delta$ 
or sin$\delta$ ($\delta$ is the transit latitude, here $\delta$ = 
90$^{\circ}$), and ``$e$'' (which is the eccentricity, the same orbital 
parameter you have heard about in class for our solar system 
planets, or in the orbit of Mercury lab, for circular orbits $e$= 1.0) 
are equal to ``1'' or ``0''.\\

First, let's remember Kepler's third law P$^{\rm 2}$ $\propto$ $a^{\rm 3}$,
where P is the orbital period, and $a$ is the semi-major axis. For Earth,
we have P = 1 yr, $a$ = 1 AU. By taking ratios, you can figure out the
orbital periods and semi-major axes of other planets in {\it our} solar system.
Here we cannot do that, and we need to use Isaac Newton's reformulation
of Kepler's third law:

\setcounter{equation}{0}
\begin{equation}
P^2 = \frac{4 \pi^2 a^3}{G(M_{star} + M_{planet})}
%\frac{4 \pi^2 a^3}{G(M_{\sun} + M_{planet})} 
\end{equation}

``G'' in this equation is the gravitational constant (G = 6.67 x 10$^{\rm -11}$
Newton-m$^{\rm 2}$/kg$^{\rm 2}$), and $\pi$ = 3.14. 

We also have to estimate the size of the planet. As detailed in Fig. 
\ref{transit}, the depth of the ``eclipse'' gives us the ratio of the radius 
of the planet to that of the star:

\begin{equation}
\frac{\Delta F}{F} = \Bigg(\frac{R_{planet}}{R_{star}}\Bigg)^2
\end{equation}

Now we have everything we need to use transits to characterize exoplanets.
We will have to re-arrange equations 1 and 2 so as to extract unknown
parameters where the other variables are known from measurements.

\subsection{Deriving Parameters from Transit Light Curves}

The orbital period of the exoplanet is the easiest
parameter to measure. In Figure \ref{kepler} is the light curve of
``Kepler 1b'', the first of the exoplanets examined by the
{\it Kepler} mission. Kepler 1b is a Hot Jupiter, so it has a deep transit. 
You can see from the figure that transits recur every 2.5 days. That is the 
orbital period of the planet. It is very easy to figure out orbital periods,
so we will not be doing that in this lab today.\\

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.5]{Exoplanets/KeplerLC.jpg}
\caption{The light curve of Kepler 1b as measured by the {\it Kepler} satellite.
The numbers on the y-axis are the total counts (how much light was measured),
while the x-axis is ``modified Julian days''. This is a system that simply
makes it easy to figure out periods of astronomical events since it is
a number that increases by 1 every day (instead of figuring out how
many days there were between June 6$^{\rm th}$ and November 3$^{\rm rd}$). 
Thus, to get an orbital period you just subtract the MJD of one event from 
the MJD of the next event.}
\label{kepler}
\end{figure}

In the following eight figures are the light curves of eight different
transiting exoplanets. Today you will be using these light curves
to determine the properties of transiting exoplanets. To help you through
this complicated process, the data for exoplanet \#8 will be worked out
at each step below. You will do the same process for one of the other seven 
transiting exoplanets. Your TA might assign one to you, or you will be left
to choose one. Towards the end of today's exercise your group will classify
both of these exoplanets. Each panel lists the orbital period of the 
exoplanets (``xxx day orbit''), ranging from 3.89 days for exoplanet \#3, to 
3.48 years for exoplanet \#2. You should be able to guess what that means 
already: one is close to its host star, the other far away. The other 
information contained in these figures is a measurement of ``t'', the total 
time of the transit (``eclipse takes xxx hours''). When working
with the equations below, all time units must be in seconds! Remember,
3600 seconds per hour, 24 hours per day, 365 days per year (there are
3.15 $\times$ 10$^{\rm 7}$ seconds per year).\\

\begin{flushleft}
{\bf Exercise \#1:}
\end{flushleft}

\begin{enumerate}

\item The first quantity we need to calculate is the size of the planet
with respect to the host star. How do we do that? Go back to Figure 
\ref{transit}. We need to measure ``$\Delta$F/F''. The data points in the
exoplanet light curves have been fit with a transit model (the solid line
fit to the data points) to make it easy to measure the {\it minimum}.
For both of the transits, take a ruler and determine the value 
on the y axis by drawing a line across the model fit to the light curve minimum.
Estimate this number as precisely as possible, then subtract this number from
1, and you get $\Delta$F/F. ({\bf 2 points})

\end{enumerate}
\begin{center}
$\Delta$F/F for transit \# \underline{\hspace{2cm}} = \underline{\hspace{5cm}}\\
\vspace{0.2in}
$\Delta$F/F for transit \hspace{0.9cm}\#8\hspace{0.9cm} = \underline{\hspace{2.0cm}0.00153\hspace{2.0cm}}\\
\end{center}

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit01.jpg}
\caption{Transiting exoplanet \#1. The vertical line in the center of the
plot simply identifies the center of the eclipse.}
\label{transit01}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit02.jpg}
\caption{Transiting exoplanet \#2.}
\label{transit02}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit03.jpg}
\caption{Transiting exoplanet \#3.}
\label{transit03}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit04.jpg}
\caption{Transiting exoplanet \#4.}
\label{transit04}
\end{figure}

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit05.jpg}
\caption{Transiting exoplanet \#5.}
\label{transit05}
\end{figure}

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit06.jpg}
\caption{Transiting exoplanet \#6.}
\label{transit06}
\end{figure}

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit07.jpg}
\caption{Transiting exoplanet \#7.}
\label{transit07}
\end{figure}

\begin{figure}[htb!]
\centering
\includegraphics[scale=0.9]{Exoplanets/transit08.jpg}
\caption{Transiting exoplanet \#8.}
\label{transit08}
\end{figure}
\clearpage
Going back to equation \#4, we have:

\begin{equation*}
\frac{\Delta F}{F} = \Bigg(\frac{R_{planet}}{R_{star}}\Bigg)^2 ~ ~ or ~ ~
R_{planet} = \Bigg(\frac{\Delta F}{F}\Bigg)^{1/2} (\times ~ R_{star})
\end{equation*}
\begin{enumerate}
\setcounter{enumi}{1}
\item Taking the square roots of the $\Delta$F/F from above, fill in the 
following blanks ({\bf 4 points}):
\end{enumerate}
\begin{center}
$R_{planet}$ for transit \# \underline{\hspace{2cm}} = \underline{\hspace{5cm}}($\times$ $R_{star}$)\\
\vspace{0.2in}
$R_{planet}$ for transit \underline{\hspace{1.15cm}\#8\hspace{0.7cm}} = \underline{\hspace{1.9cm}0.0391\hspace{2.1cm}}($\times$ $R_{star}$)\\
\end{center}

You just calculated the relative sizes of the planets to their host stars.
To turn these into real numbers, we have to know the sizes of the host 
stars. Astronomers can figure out the masses, radii, temperatures and
luminosities of stars by combining several techniques (photometry, parallax, 
spectroscopy, and interferometry). Note that stars can have dramatically
different values for their masses, radii, temperatures and luminosities,
and these directly effect the parameters derived for their exoplanets.
The data for the eight exoplanet host stars are listed in Table \ref{hostdata}. 
The values for our Sun are M$_{\odot}$ = 2 $\times$ 10$^{\rm 30}$ kg, R$_{\odot}$ = 7 $\times$ 10$^{\rm 8}$ m, L$_{\odot}$ = 4 $\times$ 10$^{\rm 26}$ Watts.

\begin{table}[htb]
\begin{center}
\caption{Exoplanet Host Star Data}
\begin{tabular}{|l|c|c|c|c|}\hline
{\bf Object} &{\bf Mass} &{\bf Radius}&{\bf Temperature}&{\bf Luminosity}\\
             & (kg) & (meters)& (K) & (Watts)\\
\hline\hline
\#1   &2.0$\times$10$^{\rm 30}$&7.00$\times$10$^{\rm 8}$&5800&4.0$\times$10$^{\rm 26}$  \\ \hline 
\#2   &1.3$\times$10$^{\rm 30}$&4.97$\times$10$^{\rm 8}$&4430&2.8$\times$10$^{\rm 25}$ \\ \hline
\#3   &2.2$\times$10$^{\rm 30}$&7.56$\times$10$^{\rm 8}$&6160&1.2$\times$10$^{\rm 27}$  \\ \hline
\#4   &2.0$\times$10$^{\rm 30}$&7.00$\times$10$^{\rm 8}$&5800&4.0$\times$10$^{\rm 26}$  \\ \hline
\#5   &1.6$\times$10$^{\rm 30}$&5.88$\times$10$^{\rm 8}$&5050&2.4$\times$10$^{\rm 26}$  \\ \hline
\#6   &2.0$\times$10$^{\rm 30}$&7.00$\times$10$^{\rm 8}$&5800&4.0$\times$10$^{\rm 26}$  \\ \hline
\#7   &1.4$\times$10$^{\rm 30}$&5.25$\times$10$^{\rm 8}$&4640&4.8$\times$10$^{\rm 25}$ \\ \hline
\#8   &1.0$\times$10$^{\rm 30}$&3.99$\times$10$^{\rm 8}$&3760&4.0$\times$10$^{\rm 24}$ \\ \hline
\end{tabular}
\label{hostdata}
\end{center}
\end{table}

\begin{enumerate}
\setcounter{enumi}{2}

\item Now that you calculated the radius of the exoplanet with respect to
the host star radius, use the data in Table \ref{hostdata} to convert
the radii of your planet into meters, and put this value in the correct
row and column in Table \ref{exodata}. ({\bf 5 points})\\
\end{enumerate}
\begin{table}
\renewcommand{\arraystretch}{2}
\begin{center}
\caption{Exoplanet Data}
\begin{tabular}{|l|c|c|c|c|c|}\hline
{\bf Object} &~ ~ ~ {\bf Radius} ~ ~ ~&~ {\bf Semi-major} ~&~ ~ ~ {\bf Mass} ~ ~ ~&{\bf Density} &{\bf Temperature}\\
             & (m) & {\bf axis} (m)& (kg) & (kg/m$^{\rm 3}$)&(K) \\
\hline\hline
\#1   &&&1.9 $\times$ 10$^{\rm 26}$ &&  \\ \hline
\#2   &&&1.9 $\times$ 10$^{\rm 28}$ &&  \\ \hline
\#3   &&&5.7 $\times$ 10$^{\rm 27}$ &&  \\ \hline
\#4   &&&6.0 $\times$ 10$^{\rm 24}$ &&  \\ \hline
\#5   &&&1.5 $\times$ 10$^{\rm 25}$ &&  \\ \hline
\#6   &&&8.0 $\times$ 10$^{\rm 26}$ &&  \\ \hline
\#7   &&&4.0 $\times$ 10$^{\rm 24}$ &&  \\ \hline
\#8   &1.6$\times$10$^{\rm 7}$&9.0$\times$10$^{\rm 9}$&5.5 $\times$ 10$^{\rm 25}$ &3205& 555 \\ \hline
\end{tabular}
\label{exodata}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
\begin{enumerate}
\setcounter{enumi}{3}
\item Astronomer Judy, and her graduate student Bob, used the spectrograph on 
the Keck telescope in Hawaii to measure the masses of your planets using the 
radial velocity technique mentioned above. So we have entered their values for 
the masses for all of the exoplanets in Table \ref{exodata}. You need to 
calculate the density of your exoplanet and enter it in the
correct places in Table \ref{exodata}. Remember that density = 
mass/volume, and the volume of all of the planets is V = 4$\pi$R$^{\rm 3}$/3,
as we know that they all must be spherical. 
({\bf 5 points})\\

\item By calculating the density, you already know something about your 
planets. Remember that the density of Jupiter is 1326 kg/m$^{\rm 3}$ and
the density of the Earth is 5514 kg/m$^{\rm 3}$. If you did the Density
lab this semester, we used the units of gm/cm$^{\rm 3}$, where water
has a density of 1.00 gm/cm$^{\rm 3}$. This is the ``cgs'' system of units.
To get from kg/m$^{\rm 3}$ to gm/cm$^{\rm 3}$, you simply divide by 1000.
Describe how the densities of your two exoplanets compare with the Earth 
and/or Jupiter. ({\bf 5 points})\\
\vspace{2.0in}

The next parameter we want to calculate is the semi-major axis ``{\it a}''.
While we now know the size and densities of our planets, we do not know
how hot or cold they are. We need to figure out how far
away they are from their host stars. To do this we re-arrange equation \#1, 
and we get this:\\

\begin{equation*}
a = \bigg(\frac{P^2G(M_{star} + M_{planet})}{4\pi^{2}}\bigg)^{1/3} =
(1.69 \times 10^{-12} P^2 M_{star})^{1/3}
\end{equation*}

\item You must use seconds for {\it P}, and kg for the mass of the
star (note: you can ignore the mass 
of the planet since it will be very small compared to the star). We
have simplified the equation by bundling G and 4$\pi ^{\rm 2}$ into a
single constant. Note that
you have to take the cube root of the quantity inside the parentheses.
We write the cube root as an exponent of ``1/3''. Ask your TA for help
on this step. Fill in the 
column for semi-major axis in Table \ref{exodata} for your exoplanet. 
({\bf 5 points})\\
\end{enumerate}

\subsection{The Habitable Zone}
	
The habitable zone is the region around a star in which the conditions are 
just right for a planet to have liquid water on its surface. Here on Earth, 
all life must have access to liquid water to survive. 
Therefore, a planet is considered ``habitable" if it has liquid water. This 
zone is also colloquially know as the ``Goldilocks Zone". \\ 

To figure out the temperature of a planet is actually harder than you
might think. We know how much energy the exoplanet host stars emit, as
that is what we call their luminosities. We also know how far away your 
exoplanets are from this energy source (the semi-major axis). 
The formula to estimate the ``equilibrium
temperature'' of an exoplanet with a semi-major axis of {\it a} around a host 
star with known parameters is:\\

\begin{equation}
T_{planet} = T_{star}(1.0 - A)^{1/4}\Bigg(\frac{R_{star}}{2a}\Bigg)^{1/2}
\end{equation}
~\\
The ``A'' in this equation is the ``Albedo'', how much of the energy 
intercepted by a planet is reflected back into space. Equation \#3 is not too 
hard to derive, but we do not have enough time to explain how it arises. You 
can ask your professor, or search Wikipedia using the term ``Planetary 
equilibrium temperature'' to find out where this comes from. The big problem 
with using this equation is that different atmospheres create different 
effects. For example, Venus reflects 67\% of the visible light from the Sun, 
yet is very hot. The Earth reflects 39\% of the visible light from the Sun
and has a comfortable climate. It is how the atmosphere ``traps heat'' that 
helps determine the surface temperature. Alternatively, a planet might not 
even have an atmosphere and could be bright or dark with no heat trapping
(for example, the Albedo of the moon is 0.11, as dark as asphalt, and
the surface is boiling hot during the day, and extremely cold at night).\\

Let's demonstrate the problem using the Earth. If we use the value of A = 0.39 
for Earth, equation \#3 would predict
a temperature of T$_{\rm Earth}$ = 247 K. But the mean temperature on the
Earth is actually T$_{\rm Earth}$ = 277 K. Thus, the atmosphere on Earth
keeps it warmer than the equilibrium temperature. This is true for just
about any planet with a significant atmosphere. To account for this effect,
let's go backwards and solve for ``A''. With R$_{\odot}$ = 
7.0 $\times$ 10$^{\rm 8}$ m, $a$ = 1.50 $\times$ 10$^{\rm 11}$ m, 
T$_{\rm Earth}$ = 277 K, and T$_{\odot}$ = 
5800 K, we find that A = 0.05. Thus, the Earth's atmosphere makes it seem
like we absorb 95\% of the energy from the Sun. We will presume this is
true for all of our planets.\\

If we assume A = 0.05, equation \#3 simplifies to: \\

\begin{equation}
T_{planet} = 0.70\Bigg(\frac{R_{star}}{a}\Bigg)^{1/2}T_{star}
\end{equation}

\begin{flushleft}
[To understand what we did here, note that (1.0 $-$ A) = 0.95. The fourth
root of 0.95 = 0.95$^{1/4}$ = 0.99 (remember the fourth root is two
successive square roots: $\sqrt{0.95}$ = 0.95$^{1/2}$ = 0.97, and 
0.97$^{1/2}$ = 0.99). We then divided 0.99 by $\sqrt{2}$ (= 1.41) to have a 
single constant out front.]\\
\end{flushleft}

\begin{enumerate}
\setcounter{enumi}{6}
\item Calculate the temperature of your exoplanet using equation
\#4 and enter it into Table \ref{exodata}. ({\bf 5 points})\\
\end{enumerate}
	
As we said, the habitable zone is the region around a star of a particular 
luminosity where water might exist in a liquid form somewhere on a planet
orbiting that star. The Earth ({\it a} = 1 AU) 
sits in the habitable zone for the Sun, while Venus is too close to the Sun
({\it a} = 0.67 AU) to be inside the habitable zone, while Mars ({\it a} = 
1.52 AU) is near the outer edge. As 
we just demonstrated, the atmosphere of a planet can radically change the 
location of the habitable zone. Mars has a very thin atmosphere, so it is
very cold there and all of its water is frozen. If Mars had the thick 
atmosphere of Venus, it would probably have abundant liquid water on its 
surface. As we noted, the mean temperature of Earth is 277 K, but
the polar regions have average temperatures well below freezing (32$^{\circ}$F
= 273 K) with
an average annual temperature at the North pole of 263 K, and 228 K at
the South pole. The equatorial regions of Earth meanwhile have average 
temperatures of 300 K. So for just about every planet there will be wide 
ranges in surface temperature, and liquid water could exist somewhere on 
that planet.\\
\begin{enumerate}
\setcounter{enumi}{7}

\item Given that your temperature estimates are not very precise, 
we will consider your planet to be in the habitable zone if its temperature is 
between 200K and 350 K. Is either of your planets in the habitable zone?
({\bf 4 points)}\\
\end{enumerate}
\vspace{0.25in}

\subsection{Classifying Your Exoplanets}

At the beginning of today's lab we described the several types of exoplanet
classes that currently exist. We now want you to classify your exoplanet
into one of these types. To help you decide, in Table \ref{solarsystem}
we list the parameters of the planets in our solar system. After you have
classified them, you will ask your TA to see ``images'' of your exoplanets
to check to see how well your classifications turned out.

\begin{table}[htb!]
\renewcommand{\arraystretch}{1.3}
\begin{center}
\caption{Solar System Data}
\begin{tabular}{|l|c|c|c|c|c|}\hline
{\bf Object} &~ ~ ~ {\bf Radius} ~ ~ ~&~ {\bf Semi-major} ~&~ ~ ~ {\bf Mass} ~ ~ ~&{\bf Density} &{\bf Temperature}\\
             & (m) & {\bf axis} (m)& (kg) & (kg/m$^{\rm 3}$)&(K) \\
\hline\hline
Mercury &2.44 $\times$ 10$^{\rm 6}$&5.79 $\times$ 10$^{\rm 10}$&3.3 $\times$ 10$^{\rm 23}$ &5427&445  \\ \hline
Venus   &6.05 $\times$ 10$^{\rm 6}$&1.08 $\times$ 10$^{\rm 11}$&4.9 $\times$ 10$^{\rm 24}$ &5243&737    \\ \hline
Earth   &6.37 $\times$ 10$^{\rm 6}$&1.49 $\times$ 10$^{\rm 11}$&5.9 $\times$ 10$^{\rm 24}$ &5514& 277 \\ \hline
Mars    &3.39 $\times$ 10$^{\rm 6}$&2.28 $\times$ 10$^{\rm 11}$&6.4 $\times$ 10$^{\rm 23}$ &3933& 210 \\ \hline
Jupiter &6.99 $\times$ 10$^{\rm 7}$&7.78 $\times$ 10$^{\rm 11}$&1.9 $\times$ 10$^{\rm 27}$ &1326&122  \\ \hline
Saturn  &6.03 $\times$ 10$^{\rm 7}$&1.43 $\times$ 10$^{\rm 12}$&5.7 $\times$ 10$^{\rm 26}$ &687& 90  \\ \hline
Uranus  &2.54 $\times$ 10$^{\rm 7}$&2.87 $\times$ 10$^{\rm 12}$&8.7 $\times$ 10$^{\rm 25}$ &1270&63  \\ \hline
Neptune &2.46 $\times$ 10$^{\rm 7}$&4.50 $\times$ 10$^{\rm 12}$&1.0 $\times$ 10$^{\rm 26}$ &1638& 50 \\ \hline
Pluto   &1.18 $\times$ 10$^{\rm 6}$&5.87 $\times$ 10$^{\rm 12}$&1.3 $\times$ 10$^{\rm 22}$ &2030& 43 \\ \hline
\end{tabular}
\label{solarsystem}
\end{center}
\end{table}

\begin{enumerate}
\setcounter{enumi}{8}
\item Compare the radii, the semi-major axes, the masses, densities and
temperatures you found for your two exoplanets to the values found in
our solar system. For example, if the radius of one of your exoplanets was
8 $\times$ 10$^{\rm 7}$, and its mass was 2.5 $\times$ 10$^{\rm 27}$
it is similar in ``size'' to Jupiter. But it could have a higher or lower
density, depending on composition, and it might be hotter than Mercury, or 
colder than Mars. Fully describe your two exoplanets. ({\bf 10 points})\\
\end{enumerate}
 ~\\


\vspace{3.0in}

As Table \ref{solarsystem} shows you, there are two main kinds of planets
in our solar system: the rocky Terrestrial planets with relatively thin
atmospheres, and the Jovian planets, which are gas giants. Planets with
high densities ($>$ 3000 kg/m$^{\rm 3}$) are probably like the Terrestrial
planets. Planets with low densities ($<$ 3000 kg/m$^{\rm 3}$) are probably
mostly gaseous or have large amounts of water (Pluto has a large fraction
of its mass in water ice).

\begin{enumerate}
\setcounter{enumi}{9}
\item Given your discussion from the previous question, and the discussion
of the types of exoplanets in the introduction, classify your
two exoplanets into one of the following categories: 1) Gas giant, 2) Hot
Jupiter, 3) Water world, 4) Exo-Earth, 5) Super-Earth, or 6) Chthonian.
What do you expect them to look like? ({\bf 10 points})\\
\vspace{3.0in}

\item Your TA has images for all eight exoplanets of this lab obtained from 
NASA's ``Exoplanet Imager'' mission that was successfully launched in 2040. 
Were your predictions correct? Yes/no. If no, what went wrong? [The TA also
has the data for all of the exoplanets to help track down any errors.] 
({\bf 10 points})\\
\vspace{2.0in}
\end{enumerate}

\clearpage
        \newpage
\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}
        \subsection{Take Home Exercise (35 points total)}
Please summarize the important concepts discussed in this lab. Your summary 
should include:
\vspace{5mm}

        \begin{itemize}
\item Discuss the different types of exoplanets and their characteristics.
                
\item What are the measurements required for you to determine the most
important parameters of an exoplanet?

\item What requirement for an exoplanet gives it the possibility of 
harboring life?
        \end{itemize}

	\vspace{5mm}
	Use complete sentences, and proofread your summary before handing in the lab.
	
	\subsection{Possible Quiz Questions}
	\begin{enumerate}
		\item What are some of the different types of exoplanets?
		\item What are some different exoplanet detection methods?
		\item What is the habitable zone?
		\end{enumerate}
		
	\vspace{5mm}
	
	\subsection{Extra Credit (ask your TA for permission before
		attempting, 5 points )}
Your TA has the data for all of the exoplanets for today's lab. With that
data, go back and answer questions \#8 and \#9 for all of the exoplanets.\\

\vspace{1.0in}
Acknowledgement: This lab was made possible using the Extrasolar Planets 
Module of the Nebraska Astronomy Applet Project.\\

~\\
\clearpage
~ \\
\newpage
%\end{document}

