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% 
% ExtrasolarPlanetsAST2011.tex
%
% Peter Newbury 
% newbury@phas.ubc.ca
% Twitter @polarisdotca
% http://blogs.ubc.ca/polarisdotca
%
% 7 July 2011
% 
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\newcommand{\mybox}{\psframe(-1ex,0)(1ex,2ex)}
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\newcommand{\foo}{1in}

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  \psline(0,0)(1,0)(1,1)(0,1)(0,0)}

\begin{document}

\begin{center}
  \huge
  Extrasolar Planets\\
\end{center}

Astronomers have discovered hundreds of planets orbiting other stars.
These planets are in solar systems beyond ours so they are called
``extrasolar'' planets. 

\medskip

\begin{tabularx}{\textwidth}[t]{@{}Xc@{}}
  
  The majority of the planets have been discovered by the {\bf radial
    velocity (Doppler) method} which detects periodic redshifts and
  blueshifts in the star's light as it (and the extrasolar planet) orbit
  about the center of mass.  Astronomers have {\bf directly} observed
  a few extrasolar planets, that is, actually seen the planet.  

  \medskip

  A growing number extrasolar planets are found by the {\bf
    transit method}.  In the transit method, astronomers take precise,
  long-term observations of the brightness (or ``intensity'') of a star
  and create a {\bf light curve} for the star.  For most stars, the
  brightness remains constant and the light curve is flat.  For some
  stars, there is a regular dip in the light curve when the extrasolar
  planet passes between us and the star (``transits the star'') blocking
  some of the star light travelling to Earth.  The shape of this dip
  depends characteristics of the extrasolar planet and its orbit.

  &

  % Extrasolar Planet figure
  %\makebox[7cm]{%
  %  \rput[t](0,2ex){\psframebox[framesep=0.5pt,linewidth=0.25pt]{%
  %      \includegraphics[width=7cm]{15_13Figureb}}}}
  \renewcommand{\foo}{7.8cm}
  \begin{tabular}[t]{@{}p{\foo}@{}}
    \ \\[-2ex]
    \includegraphics[width=\foo]{Kepler11_NASA}\\
    Artist's impression of a star with transiting extrasolar planets. (Credit:
    NASA/Tim Pyle)
  \end{tabular}
\end{tabularx}

In this tutorial, you'll explore the connections between light curves
and extrasolar planets and learn how to decode the light curve.  Then
you'll examine the light curve of a real star and discover the
characteristics of the planet HD~209458b, the first
transiting extrasolar planet ever found.


\vfill

{\Large Part 1: Observations}  \ \ Write down any patterns you observe
during the demonstration. 

\bigskip

%\renewcommand{\arraystretch}{1.25}
\renewcommand{\arraystretch}{1.2}

\begin{tabularx}{\textwidth}{|X|X|}\hline
  When the demonstrator... & ...the light curve \\\hline
  {\em puts the ball in front of the light} &
  {\em dips} \\[25em]\hline  
\end{tabularx}

\clearpage

{\Large Part 2: Characteristics of Extrasolar Planets and Light Curves}

\bigskip

Exactly how planets in a solar system orbit the star and block its light
can be very complicated.  A very good approximation, though, depends on
only {\bf two characteristics}:
\begin{itemize}
\item the {\bf diameter of the planet} relative to the diameter of the
  star.  For example, Earth is $13\,000$~km in diameter, compared
  to the Sun's $1\,400\,000$~km, so Earth's diameter is $13\,000 /
  1\,400\,000 = 0.009$ of the Sun's. 
\item the {\bf orbital period of the planet}: how long it takes the
  planet to travel once around its sun.  Earth's orbital period is 1~year.
\end{itemize}

Each pair of graphs below shows the light curves a star with two planets and the
dips that occur when one of the planets transits the star.

\newcommand{\lightcurve}[2]{%
  \psset{xunit=9.5mm,yunit=4mm,arrowinset=0}
  \pspicture(-0.5,93)(8.5,102)
  % frame
  \psframe[linewidth=0.5pt](-0.5,95)(8.5,101)
  % time ticks
  \multirput(0,95)(1,0){9}{\psline[linewidth=0.5pt](0,0)(0,1ex)}
  \rput[t]{U}(4,94.5){Time}
  % intensity ticks, both sides
  \psline[linewidth=0.5pt](-0.5,100)(8.5,100)
  \multirput(-0.5,96)(0,1){5}{\psline[linewidth=0.5pt](0,0)(1ex,0)}
  \multirput(8.5,96)(0,1){5}{\psline[linewidth=0.5pt](0,0)(-1ex,0)}
  \rput{L}(-1,98){Intensity}
  % title
  \rput[b]{U}(4,101.25){Planet\ #1}
  % the lightcurve
  \psset{linewidth=1pt}
  #2
  \endpspicture}

\renewcommand{\foo}{20mm}

\medskip
%
% Case 1: change planet's diameter
%
\begin{tabular}{c@{\hspace{20mm}}c}
  % Planet A
  \lightcurve{A}{%
    % depth 1 dip at t=2
    \psline(-0.5,100)(1,100)
    \psbezier(1,100)(1.5,100)(1.5,99)(2,99)
    \psbezier(2,99)(2.5,99)(2.5,100)(3,100)
    \psline(3,100)(8.5,100)
  } % end of planet A
  &
  % Planet B
  \lightcurve{B}{%
    % depth 4 dip at t=6
    \psline(-0.5,100)(5,100)
    \psbezier(5,100)(5.5,100)(5.5,96)(6,96)
    \psbezier(6,96)(6.5,96)(6.5,100)(7,100)
    \psline(7,100)(8.5,100)
  } % end of planet B
\end{tabular}

What {\bf feature} of the light curve changed? {\bf How} did it change?

\vspace{\foo}

What {\bf characteristic} of planets is different: diameter or orbital
period? {\bf How} is it different?

\vspace{\foo}

\vfill

%
% Case 2: change orbital orbital
%
\begin{tabular}{c@{\hspace{20mm}}c}
  % Planet C
  \lightcurve{C}{%
    % depth 2 dip w period 2
    %\psline(-0.5,100)(-0.5,100)
    \psbezier(-0.5,100)(-0.25,100)(-0.25,98)(0,98)
    \psbezier(0,98)(0.25,98)(0.25,100)(0.5,100)
    \psline(0.5,100)(1.5,100)
    % @2
    \psbezier(1.5,100)(1.75,100)(1.75,98)(2,98)
    \psbezier(2,98)(2.25,98)(2.25,100)(2.5,100)
    \psline(2.5,100)(3.5,100)
    % @ 4
    \psbezier(3.5,100)(3.75,100)(3.75,98)(4,98)
    \psbezier(4,98)(4.25,98)(4.25,100)(4.5,100)
    \psline(4.5,100)(5.5,100)
    % @ 6
    \psbezier(5.5,100)(5.75,100)(5.75,98)(6,98)
    \psbezier(6,98)(6.25,98)(6.25,100)(6.5,100)
    \psline(6.5,100)(7.5,100)
    % @ 8
    \psbezier(7.5,100)(7.75,100)(7.75,98)(8,98)
    \psbezier(8,98)(8.25,98)(8.25,100)(8.5,100)
    %\psline(2.5,100)(3.5,100)
  } % end of planet C
  &
  % Planet D: depth 2 dip with period 4
  % a^3/p^2 means a=4^(1/3)~1.59 of C's
  % speed  (2 pi a)/p becomes (2 pi 1.59 a)/4 = 0.79 C's
  % transit time is 1/0.79 ~ 1.26 longer: make dips 1.25 wider
  \lightcurve{D}{%
    % depth 2 dip with t=4 at t=2
    \psline(-0.5,100)(1.375,100)
    \psbezier(1.375,100)(1.6875,100)(1.6875,98)(2,98)
    \psbezier(2,98)(2.3125,98)(2.3125,100)(2.625,100)
    \psline(2.625,100)(5.375,100)
    % 
    \psbezier(5.375,100)(5.6875,100)(5.6875,98)(6,98)
    \psbezier(6,98)(6.3125,98)(6.3125,100)(6.625,100)
    \psline(6.625,100)(8.5,100)
  } % end of planet D
\end{tabular}

What {\bf feature} of the light curve changed? {\bf How} did it change?

\vspace{\foo}

What {\bf characteristic} of planets is different: diameter or orbital
period? {\bf How} is it different?

\vspace{\foo}


\clearpage

{\Large Part 3: Decoding the Light Curve}

\bigskip

The characteristics of a transiting extrasolar planet are hidden in the
shape of the star's light curve:

\medskip
{\bf Orbital Period}\ \ The orbital period $P$ of the planet is simply
the length of time between the transits, which appear as dips in the
light curve.  To measure this length of time, use some particular
feature of the dip, like its beginning, middle or end.  Even better,
measure the time between several dips and divide.


\medskip
\begin{tabularx}{\textwidth}{@{}Xp{3cm}}
  {\bf Diameter of the Planet} \ \ The light curve dips when the planet
  travels in front of the star and blocks some of the star light from
  the Earth.  The fraction by which the intensity dips is the ratio of
  the area of the disk of the planet compared to the area of the disk of
  the star.

  \medskip
  If the planet has diameter $d$ and the star has diameter $D$,
  the drop in intensity is
  \[
  \mathrm{drop\ in\ intensity\ } \Delta I =  
  \displaystyle\frac{\mathrm{area\ of\ planet's\ disk}}{%
    \mathrm{area\ of\ star's\ disk}} =
  \displaystyle\frac{\pi (\frac{d}{2})^2}{\pi (\frac{D}{2})^2} =
  \displaystyle\left(\frac{d}{D}\right)^2
  \ \ \ \ \mathrm{so}\ \ \ \ 
  \displaystyle\frac{d}{D} = \sqrt{\Delta I}
  \]
  &
  \psset{unit=12mm,linewidth=0.5pt,arrowinset=0}
  \rput[tl](0,2ex){%
    \pspicture*(-1.2,-1.6)(1.2,1.2)
    \psframe(-1.2,-1.59)(1.2,1.2)
    % star
    \pscircle[linewidth=1pt,dimen=middle](0,0){1}
    % transit path
    %\psline[linewidth=0.25pt,linestyle=dashed](-2,-0.2)(2,-0.2)
    %\psline[linewidth=0.25pt,linestyle=dashed](-2,0.2)(2,0.2)
    % multiplanets
    %\multido{\nx=-1.2+0.4}{8}{%
    %  \pscircle[linewidth=0.5pt,linestyle=dashed](\nx,0){0.2}}
    \multido{\nx=-1.2+0.1}{8}{%
      \pscircle[linewidth=0.25pt,fillstyle=none,fillcolor=lightgray](\nx,0){0.2}}
    % planet
    \pscircle[dimen=outer,linewidth=1pt,fillstyle=solid,fillcolor=gray](-0.40,0){0.2}
    % measurement lines
    \psline{<->}(-0.60,0.30)(-0.20,0.30)
    \uput[90](-0.40,0.30){$d$}
    \psline{<->}(-0.95,-1.1)(0.95,-1.1)
    \uput[-90](0,-1.1){$D$}
    \endpspicture}\\[-3ex]
\end{tabularx}

\vspace{1cm}

The first transiting extrasolar planet was found orbiting a star
named HD~209458. The star and planet are 150~light years
away in the constellation, Pegasus.  The long, light curve poster
contains more than $70\,000$ measurements of the intensity of the star,
collected with the {\em MOST} (Microvariability 
and Oscillations of STars) space telescope. {\em MOST} is operated by the
University of British Columbia.

\medskip
Examine the light curve and take two measurements listed below.  Then find the
characteristics that describe the extrasolar planet HD~209458b orbiting
the star.

\bigskip
\renewcommand{\foo}{3mm}
\renewcommand{\arraystretch}{2}
\begin{tabularx}{\textwidth}{|X|X|}\hline
  \multicolumn{1}{>{\columncolor[gray]{0.8}}c|}{Orbital period} & 
  \multicolumn{1}{>{\columncolor[gray]{0.8}}c|}{Planet diameter}\\\hline
  %
  Measure the time between dips & 
  Measure the depth of the dip  \\[\foo]
  %
  \ \ $P=$ &
  \ \ $\Delta I=$ \\[\foo]\hline
  %
  write the orbital period in days &
  convert \% dip to a decimal (for example, $1\%=0.01$) \\[\foo]
  %
  \ \ $P=$ &
  \ \ $\Delta I=$ \\[\foo]
  %
  and years &
  find the ratio of diameters  \\[\foo]
  %
  \ \ $P=$ &
  \ \ $\displaystyle\frac{d}{D} = \sqrt{\Delta I} =$  \\[\foo]
  %
  & 
  HD~209458 has diameter $D=1\,400\,000$~km, the same size as our Sun. Find diameter
  $d$ in km \\[\foo] 
  & 
  \ \ $d=$ \\[\foo]\hline
  \multicolumn{2}{|>{\columncolor[gray]{0.8}}c|}{%
      After you've found planet's the period and diameter, ask your TA for
      Part 4: Questions}
  \\\hline

\end{tabularx}

\clearpage

\ \vspace{-8ex}

\renewcommand{\arraystretch}{1}
\begin{tabularx}{\textwidth}{@{}rXrp{3cm}@{}}
  Name & & ID No. & \\\cline{2-2}\cline{4-4}
\end{tabularx}

\bigskip
\bigskip

{\Large Part 4: Questions}\ \ \ Please hand in this worksheet when you
are finished.

\medskip
\newcommand{\qfoo}{30mm}

\begin{enumerate}
\item The star HD~209458 has the same mass as our Sun, so according to
  Kepler's Law, $a^3 = P^2$ where $a$ is extrasolar planet's semi-major axis (in
  AU) and $P$ is its orbital period (in years). Use Kepler's Law and
  your results from Part~3 to complete this Table:

  \renewcommand{\arraystretch}{1.5}
  \renewcommand{\foo}{32.5mm}
  \begin{tabular}{|l|c|c|c|}\hline
    \rowcolor[gray]{0.8}
    Planet & Period $P$ & Semi-major axis $a$ & Planet's diameter $d$ \\\hline
    HD~209458b & \makebox[\foo]{} & \makebox[\foo]{} & 
    \makebox[\foo]{} \\
    Mercury & $0.24$~years (88 days) & $0.39$ AU & $4880$~km \\    
    Jupiter & $11.86$ years & $5.2$ AU & $140\,000$~km \\
    \hline
  \end{tabular}


\item Two students are discussing their answers to Question~1:

  \begin{tabularx}{0.9 \textwidth}{@{}lX}
    {\bf Student 1:} & 
    Look at the size of the planet HD~209458b: it's a lot like Jupiter. \\
    {\bf Student 2:} & Yeah, but look at the period and semi-major axis:
    it's more like Mercury.
  \end{tabularx}

  Do you agree or disagree with either or both of these students?
  Explain your reasoning.

  \vfill
  
\item The Kepler spacecraft uses the transit method to look for
  Earth-sized, rocky planets that orbit stars like our Sun. Astronomers
  are excited to find planets at the right distance to have liquid
  water: not too far from the star where water is frozen and not too
  close to the star where it's so hot, water is vaporized. It's called
  the ``habitable'' or ``Goldilocks'' zone.
  
  \medskip
  Kepler scientists wait until they see the same dip in a star's
  lightcurve 3 times before they conclude they have found a planet.  Why
  does it take nearly 3 years to collect these observations?
  
  \vfill

\item This light curve shows the dips in the brightness of the star that
  has 2 transiting extrasolar planets.  The table gives possible diameters and
  periods of the two planets. Which choice A--E can produce the dips
  shown in the graph? 

  \medskip
  \begin{tabularx}{0.95\textwidth}{Xr@{}}
    %
    % light curve
    %
    \psset{xunit=5.5mm,yunit=5.5mm,linewidth=0.5pt}
    \pspicture(-1,93.5)(13,101)
    % x axis (No ticks on frame)
    \multirput(1,95.5)(1,0){12}{\psline[linewidth=0.5pt](0,0)(0,1ex)}
    \multido{\ix=0+1}{14}{\uput[-90](\ix,95.3){\small \ix}}
    % y axis
    \multirput(0,96)(0,1){5}{\psline[linewidth=0.5pt](0,0)(1ex,0)}
    \multirput(13,96)(0,1){5}{\psline[linewidth=0.5pt](0,0)(-1ex,0)}
    \multido{\iy=96+1}{5}{\uput[180](-2mm,\iy){\small \iy\%}}
    \psframe(0,95.5)(13,101)
    % 100% line
    \psline[linewidth=0.5pt](0,100)(13,100)
    \psset{linewidth=1pt}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % lightcurve for planet A
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % @ 2
    \psbezier(1.75,100)(2.0,100)(1.75,99)(2,99)
    \psbezier(2,99)(2.25,99)(2,100)(2.25,100)
    % @ 6
    \psbezier(5.75,100)(6.0,100)(5.75,99)(6,99)
    \psbezier(6,99)(6.25,99)(6,100)(6.25,100)
    % @ 10
    \psbezier(9.75,100)(10.0,100)(9.75,99)(10,99)
    \psbezier(10,99)(10.25,99)(10,100)(10.25,100)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % lightcurve for planet B
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % @ 5
    \psbezier(4.5,100)(5.0,100)(4.5,96)(5,96)
    \psbezier(5,96)(5.5,96)(5,100)(5.5,100)
    % @ 12
    \psbezier(11.5,100)(12.0,100)(11.5,96)(12,96)
    \psbezier(12,96)(12.5,96)(12,100)(12.5,100)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % lines at 100% between dips
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \psline(0,100)(1.75,100)
    \psline(2.25,100)(4.5,100)
    \psline(5.5,100)(5.75,100)
    \psline(6.25,100)(9.75,100)
    \psline(10.25,100)(11.5,100)
    \psline(12.5,100)(13,100)
    %labels
    \rput[t]{L}(-3,98){Intensity}
    \rput[B]{U}(6.5,93.5){Time (days)}
    \endpspicture
    &
    %
    % table of choices
    %
    \renewcommand{\foo}{20mm}
    \renewcommand{\arraystretch}{1.25}
    \begin{tabular}[b]{c|cc|cc|}\cline{2-5}
      & \multicolumn{2}{c|}{Planet 1} & \multicolumn{2}{c|}{Planet 2} \\[-1ex]
      & period & diameter & period & diameter \\[-1ex]
      Choice & (days) & (km) & (days) & (km) \\\hline
      A & 4  & $400\,000$ & 7 & $100\,000$ \\
      B & 2  &  $50\,000$ & 3 & $200\,000$ \\
      C & 4  & $100\,000$ & 7 & $200\,000$ \\
      D & 2  & $400\,000$ & 3 & $200\,000$ \\
      E & 4  & $100\,000$ & 7 & $400\,000$ \\\hline
    \end{tabular}
  \end{tabularx}

\end{enumerate}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% extra questions that were part of a post-test
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

\begin{enumerate}

\item The light curve of a star with 2 transiting extrasolar planets, $a$
  and $b$, is shown below.  What can you tell about the planets?

  \savedata{\lightcurve}[{%
      {0,100},
      {1,100},{1,99.0},{1.12,99.0},{1.12,100},
      {2,100},{2,99.4},{2.15,99.4},{2.15,100},
      {4,100},{4,99.0},{4.12,99.0},{4.12,100},
      {7,100},{7,99.0},{7.12,99.0},{7.12,100},
      {9,100},{9,99.4},{9.15,99.4},{9.15,100},
      {10,100},{10,99.0},{10.12,99.0},{10.12,100},
      {11,100}}] % end \lightcurve

  \medskip

  \begin{center}
    % lightcurve
    \psset{xunit=10mm,yunit=10mm,linewidth=0.5pt}
    \pspicture(-1.5,97)(11,100.5)
    % axis
    \multido{\ix=0+1}{12}{%
      \psline(\ix,97.9)(\ix,98.0)
      \uput[-90](\ix,97.9){\textsf{\ix}}}
    \multido{\iy=98+1}{3}{%
      \psline(-2mm,\iy)(0,\iy)
      \uput[180](-2mm,\iy){\textsf{\iy\%}}}
    \psframe[dimen=middle](0,98)(11,100.5)
    % lightcurve
    \dataplot[plotstyle=line,showpoints=false,linewidth=1pt,linearc=0.05]{\lightcurve}
    %labels
    \rput[t]{L}(-2,99.25){\textsf{Intensity}}
    \rput[b]{U}(5.5,97){\textsf{Time (days)}}
    \uput[-90](1.06,99){$a$}
    \uput[-90](2.075,99.4){$b$}
    % schematic of transits (should be at x=12, but get rid of it by
    % putting it at x=112)
    \rput[tl]{U}(112,100.5){%
      \psset{linewidth=0.5pt,unit=10mm}
      \pspicture*(-1.25,-1.25)(1.25,1.25)
      \psframe[dimen=outer](-1.25,-1.25)(1.25,1.25)
      % star
      \pscircle[linewidth=1pt](0,0){1}
      % upper transit of bigger planet
      \multido{\nx=-1.30+0.20}{30}{
        \pscircle(\nx,0.5){0.1}}
      \pscircle*(0.3,0.5){0.1}
      % lower transit of smaller planet
      \multido{\nx=-1.35+0.15}{30}{
        \pscircle(\nx,-0.5){0.075}}
      \pscircle*(-0.45,-0.5){0.075}
      %
      \endpspicture} % end \rput
    \endpspicture
  \end{center}
    
  \begin{tabularx}{0.95\textwidth}{cX}
    \checkbox\ \  & planet $b$ is bigger than planet $a$ and has a
    longer orbital period \\ 
    \checkbox\ \  & planet $b$ is smaller than planet $a$ and has the
    same orbital period \\ 
    \checkbox\ \  & planet $b$ is bigger than planet $a$ and has a
    shorter orbital period \\ 
    \checkbox\ \  & planet $b$ is smaller than planet $a$ and has a
    longer orbital period \\ 
    \checkbox\ \  & planet $b$ is the same size as planet $a$ and has a
    shorter orbital period \\ 
  \end{tabularx}


\item Can you think of situations where a star has an extrasolar planet
  but it cannot be detected by the transit method? 


  \vfill

  What's necessary for the transit method to work?

  \vfill

\end{enumerate}

\end{document}