\setcounter{figure}{0}
\setcounter{table}{0}
\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}

\begin{Large}
\section{\bf Tools for Success in ASTR 105G}
\end{Large}

\subsection{Introduction}

Astronomy is a physical science. Just like biology, chemistry, geology, and
physics, astronomers collect data, analyze that data, attempt to understand
the object/subject they are looking at, and submit their results for 
publication. Along the way astronomers use all of the mathematical techniques
and physics necessary to understand the objects they examine. Thus, just
like any other science, a large number of mathematical tools and concepts
are needed to perform astronomical research. In today's introductory lab,
you will review and learn some of the most basic concepts necessary 
to enable you to successfully complete the various laboratory exercises
you will encounter during this semester. When needed, the weekly laboratory
exercise you are performing will refer back to the examples in this 
introduction---so keep the completed examples you will do today with you at all 
times during the semester to use as a reference when you run into these
exercises later this semester (in fact, on some occasions your TA might
have you redo one of the sections of this lab for review purposes).

\subsection{The Metric System}

Like all other scientists, astronomers use the metric system. The
metric system is based on powers of 10, and has a set of measurement units
analogous to the English system we use in everyday life here in the US. In the
metric system the main unit of length (or distance) is the {\it meter}, the unit of mass is the {\it kilogram}, and the unit of liquid volume is the {\it liter.} A meter is approximately 40 inches, or about 4" longer than the yard. Thus, 100 meters is about 111 yards. A liter is slightly larger than a quart (1.0 liter = 1.101 qt). On the Earth's surface, a kilogram = 2.2 pounds.\\
\\
As you have almost certainly learned, the metric system uses prefixes to change
scale. For example, one thousand meters is one ``kilometer.'' One thousandth
of a meter is a ``millimeter.'' The prefixes that you will encounter in
this class are listed in Table~\ref{prefixes-tab}.\\

\begin{table}[htbp]
\begin{center}
\caption{Metric System Prefixes}
\begin{tabular}{|c|c|c|}
\hline
{\bf Prefix Name} & {\bf Prefix Symbol} & {\bf Prefix Value}\\
\hline
Giga	&	G	& 1,000,000,000 (one billion)\\
Mega	&	M	& 1,000,000 (one million) \\
kilo	&	k	& 1,000 (one thousand) \\\
centi	&	c	& 0.01 (one hundreth) \\
milli	&	m	& 0.001 (one thousandth) \\
micro	&	$\mu$	& 0.0000001 (one millionth) \\
nano    &	n	& 0.0000000001 (one billionth)\\
\hline
\end{tabular}
\end{center}
\label{prefixes-tab}
\end{table}
\noindent In the metric system, 3,600 meters is equal to 3.6 kilometers; 0.8 meter is
equal to 80 centimeters, which in turn equals 800 millimeters, etc. In the lab
exercises this semester we will encounter a large range in sizes and
distances. For example, you will measure the sizes of some objects/things in 
class in millimeters, talk about the wavelength of light in nanometers, and measure the sizes of features on planets that are larger than 1,000 kilometers.

\subsection{Beyond the Metric System}
When we talk about the sizes or distances to objects beyond
the surface of the Earth, we
begin to encounter very large numbers. For example, the average
distance from the Earth to the Moon is 384,000,000 meters or 384,000 kilometers
(km). The distances found in astronomy are usually so large that we
have to switch to a unit of measurement that is much larger than the
meter, or even the kilometer. In and around the solar
system, astronomers use ``Astronomical Units.'' An Astronomical Unit is the 
mean (average) distance
between the Earth and the Sun. One Astronomical Unit (AU) = 149,600,000 km.
For example, Jupiter is about 5 AU from the Sun, while Pluto's average
distance from the Sun is 39 AU. With this change in units, it is easy
to talk about the distance to other planets. It is more convenient to say that
Saturn is 9.54 AU away than it is to say that Saturn is 1,427,184,000 km from
Earth. 

\subsection{Changing Units and Scale Conversion}

Changing units (like those in the previous paragraph) and/or scale conversion 
is something you must master during this semester.  You already do this in your everyday life whether you know it or not (for example, if you travel to Mexico and you want to pay for a Coke in pesos), so {\bf do not panic!}  Let's look at some examples ({\bf 2 points each}):\\
\begin{enumerate}
\item Convert 34 meters into centimeters:\\
\vspace{0.2in}\\
Answer: Since one meter = 100 centimeters, 34 meters = 3,400 centimeters.\\
\vspace{0.15in}
\item Convert 34 kilometers into meters:\\
\vspace{0.4in}
\item If one meter equals 40 inches, how many meters are there in 400 inches?\\
\vspace{0.3in}
\item How many centimeters are there in 400 inches?\\
\vspace{0.3in}
\item In August 2003, Mars made its closest approach to Earth for the next 50,000 years.  At that time, it was only about .373 AU away from Earth.  How many km is this?\\
\vspace{0.3in}
\end{enumerate}

\subsubsection{Map Exercises}

\noindent One technique that you will use this semester involves measuring a photograph or image with a ruler, and converting the measured number into a real unit of size (or distance). One example of this technique is reading a road map.  Figure~\ref{map-fig} shows a map of the state of New Mexico. Down at the bottom left hand corner of the map is a scale in both miles and kilometers.\\

\begin{figure}[htbp]
\centerline{{\includegraphics[width=16cm]{nmmap.jpg}}}
\caption{Map of New Mexico.}
\label{map-fig}
\end{figure}

\noindent Use a ruler to determine ({\bf 2 points each}):\\
\begin{enumerate}
\setcounter{enumi}{5}
\item How many kilometers is it from Las Cruces to Albuquerque?\\
\vspace{0.3in}
\item What is the distance in miles from the border with Arizona to the border
with Texas if you were to drive along I-40?\\
\vspace{0.3in}
\item If you were to drive 100 km/hr (kph), how long would it take you to
go from Las Cruces to Albuquerque?\\
\vspace{0.3in}
\item If one mile = 1.6 km, how many miles per hour (mph) is 100 kph?\\
\vspace{0.3in}
\end{enumerate}

\subsection{Squares, Square Roots, and Exponents}

In several of the labs this semester you will encounter squares, cubes, and 
square roots. Let us briefly review what is meant by such terms as squares,
cubes, square roots and exponents. The square of a number is simply that
number times itself: 3 $\times$ 3 = 3$^{\rm 2}$ = 9. The {\it exponent} is the
little number ``2'' above the three. 5$^{\rm 2}$ = 5 $\times$ 5 = 25. The
exponent tells you how many times to multiply that number by itself: 
8$^{\rm 4}$ =  8 $\times$ 8 $\times$ 8 $\times$ 8 = 4096. The square of a
number simply means the exponent is 2 (three squared = 3$^{\rm 2}$), and the 
cube of a number means the exponent is three (four cubed = 4$^{\rm 3}$).
Here are some examples:\\

\begin{itemize}
\item 7$^{\rm 2}$ = 7 $\times$ 7 = 49\\
\item 7$^{\rm 5}$ = 7 $\times$ 7 $\times$ 7 $\times$ 7 $\times$ 7 = 16,807\\
\item The cube of 9 (or ``9 cubed'') = 9$^{\rm 3}$ = 9 $\times$ 9 $\times$ 9 = 729\\
\item The exponent of 12$^{\rm 16}$ is 16\\
\item 2.56$^{\rm 3}$ = 2.56 $\times$ 2.56 $\times$ 2.56 = 16.777\\
\end{itemize}
{\bf Your turn (2 points each):}\\
\begin{enumerate}
\setcounter{enumi}{9}
\item 6$^{\rm 3}$ = \\
\vspace{0.2in}
\item 4$^{\rm 4}$ = \\
\vspace{0.2in}
\item 3.1$^{\rm 2}$ = \\
\vspace{0.2in}
\end{enumerate}

\noindent The concept of a square root is fairly easy to understand, but
is much harder to calculate (we usually have to use a calculator).
The square root of a $number$ is that number whose square is the 
$number$: the square root of 4 = 2 because
2 $\times$ 2 = 4. The square root of 9 is 3 (9 = 3 $\times$ 3). The 
mathematical operation of a square root is usually represented by the
symbol ``$\surd$~'', as in $\sqrt{9}$ = 3. But mathematicians also represent
square roots using a $fractional$ exponent of one half: 9$^{1/2}$ = 3.
Likewise, the cube root of a number is represented as 27$^{1/3}$ = 3
(3 $\times$ 3 $\times$ 3 = 27). The fourth root is written as 
16$^{1/4}$ (= 2), and so on.  Here are some example problems:

\begin{itemize}
\item $\sqrt{100}$ = 10\\
\item 10.5$^{\rm 3}$ = 10.5 $\times$ 10.5 $\times$ 10.5 = 1157.625\\
\item Verify that the square root of 17 ($\sqrt{17}$= 17$^{1/2}$) = 4.123\\
\end{itemize}

\subsection{Scientific Notation}

The range in numbers encountered in Astronomy is enormous: from the size
of subatomic particles, to the size of the entire universe. You are certainly
comfortable with numbers like ten, one hundred, three thousand, ten million,
a billion, or even a trillion. But what about a number like one million
trillion? Or, four thousand one hundred and fifty six million billion? Such
numbers are too cumbersome to handle with words. Scientists use something
called ``Scientific Notation'' as a short hand method to represent very large
and very small numbers.
The system of scientific notation is based on the number 10. For example,
the number 100 = 10 $\times$ 10 = 10$^{\rm 2}$. In scientific notation
the number 100 is written as 1.0 $\times$ 10$^{\rm 2}$. Here are some
additional examples:\\
\begin{itemize}
\item  Ten = 10 = 1 $\times$ 10 = 1.0 $\times$ 10$^{\rm 1}$
\item One hundred = 100 = 10 $\times$ 10 = 10$^{\rm 2}$ = 1.0 $\times$ 10$^{\rm 2}$
\item One thousand = 1,000 = 10 $\times$ 10 $\times$ 10 = 10$^{\rm 3}$ = 1.0 $\times$ 10$^{\rm 3}$
\item One million = 1,000,000 = 10 $\times$ 10 $\times$ 10 $\times$ 10 $\times$ 10 $\times$ 10 = 10$^{\rm 6}$ = 1.0 $\times$10$^{\rm 6}$\\
\end{itemize}

\noindent Ok, so writing powers of ten is easy, but how do we write 6,563 in scientific 
notation? 6,563 = 6563.0 = 6.563 $\times$ 10$^{\rm 3}$. To figure out the 
exponent on the power of ten, we simply count the numbers to the {\it left} of 
the decimal point, but do not include the left-most number. Here are some more examples:\\
\begin{itemize}
\item 1,216 = 1216.0 = 1.216 $\times$ 10$^{\rm 3}$\\
\item 8,735,000 = 8735000.0 = 8.735000 $\times$ 10$^{\rm 6}$\\
\item 1,345,999,123,456 = 1345999123456.0 = 1.345999123456 $\times$ 10$^{\rm 12}$ $\approx$ 1.346 $\times$ 10$^{\rm 12}$\\
\end{itemize}
\vspace{0.1in}

\noindent Note that in the last example above, we were able to eliminate a lot of the ``unnecessary'' digits in that very large number.  While 1.345999123456 $\times$ 10$^{\rm 12}$ is technically correct as the scientific notation representation of the number 1,345,999,123,456, we do not need to keep {\bf all} of the digits to the right of the decimal place.  We can keep just a few, and approximate that number as 1.346 $\times$ 10$^{\rm 12}$.\\

\noindent {\bf Your turn! Work the following examples (2 points each):}\\
\begin{enumerate}
\setcounter{enumi}{12}
\item 121 = 121.0 = \\
\vspace{0.1in}
\item 735,000 = \\
\vspace{0.1in}
\item 999,563,982 = \\
\end{enumerate}

\noindent Now comes the sometimes confusing issue: writing very small numbers. First, 
lets look at powers of 10, but this time in fractional form. The number 0.1 = 
$\frac{1}{10}$.  In scientific notation we would write this as 1 $\times$ 10$^{\rm -1}$. 
The negative number in the exponent is the way we write the fraction $\frac{1}{10}$.
How about 0.001? We can rewrite 0.001 as $\frac{1}{10}$ $\times$ $\frac{1}{10}$ $\times$ $\frac{1}{10}$ = 
0.001 = 1 $\times$ 10$^{\rm -3}$. Do you see where the exponent comes from?
Starting at the decimal point, we simply count over to the {\it right} of the first 
digit that isn't zero to determine the exponent. Here are some examples:

\begin{itemize}
\item 0.121 = 1.21 $\times$ 10$^{\rm -1}$ \\
\item 0.000735 = 7.35 $\times$ 10$^{\rm -4}$ \\
\item 0.0000099902 = 9.9902 $\times$ 10$^{\rm -6}$ \\
\end{itemize}

\noindent {\bf Your turn (2 points each):}\\
\begin{enumerate}
\setcounter{enumi}{15}
\item 0.0121 =    \\
\item 0.0000735 =    \\
\item 0.0000000999 =    \\
\item $-$0.121 = \\
\end{enumerate}

\noindent There is one issue we haven't dealt with, and that is {\it when} to write numbers
in scientific notation. It is kind of silly to write the number 23.7 as 
2.37 $\times$ 10$^{\rm 1}$, or 0.5 as 5.0 $\times$ 10$^{\rm -1}$. You use
scientific notation when it is a more compact way to write a number to insure
that its value is quickly and easily communicated to someone else. For example,
if you tell someone
the answer for some measurement is 0.0033 meter, the person receiving that
information has to count over the zeros to figure out what that means.
It is better to say that the measurement was 3.3 $\times$ 10$^{\rm -3}$ meter.
But telling someone the answer is 215 kg, is much easier than saying
2.15 $\times$ 10$^{\rm 2}$ kg.  It is common practice that numbers bigger
than 10,000 or smaller than 0.01 are best written in scientific notation.

%How do we multiply and divide two numbers in Scientific Notation? It is a three step process: 1) multiply (divide) the numbers out front, 2) add (subtract) the exponents, and 3) reconstruct the number in Scientific Notation. It is easier to just show some examples:

%\begin{flushleft}
%(2 $\times$ 10$^{\rm 4}$) $\times$ (3 $\times$ 10$^{\rm 5}$) = (2 $\times$ 3) 
%$\times$ 10$^{\rm (4 + 5)}$ = 6 $\times$ 10$^{\rm 9}$\\
%\vspace{0.3in}
%(2.00 $\times$ 10$^{\rm 4}$) $\times$ (3.15 $\times$ 10$^{\rm 7}$) = (2.00 $\times$ 3.15) 
%$\times$ 10$^{\rm (4 + 7)}$ = 6.30 $\times$ 10$^{\rm 11}$\\
%\vspace{0.3in}
%(2 $\times$ 10$^{\rm 4}$) $\times$ (6 $\times$ 10$^{\rm 5}$) = (2 $\times$ 6) 
%$\times$ 10$^{\rm (4 + 5)}$ = 12 $\times$ 10$^{\rm 9}$ = 1.2 $\times$ 10$^{\rm 10}$\\
%\vspace{0.3in}
%(6 $\times$ 10$^{\rm 4}$) $\div$ (3 $\times$ 10$^{\rm 8}$) = (6 $\div$ 3) 
%$\times$ 10$^{\rm (4 - 8)}$ = 2 $\times$ 10$^{\rm -4}$\\
%\vspace{0.3in}
%(3.0 $\times$ 10$^{\rm 4}$) $\div$ (6.0 $\times$ 10$^{\rm 8}$) = (3.0 $\div$ 6.0) 
%$\times$ 10$^{\rm (4 - 8)}$ = 0.5 $\times$ 10$^{\rm -4}$ = 5.0 $\times$ 10$^{\rm -5}$\\
%\vspace{0.3in}
%Your turn:\\
%\vspace{0.3in}
%(6 $\times$ 10$^{\rm 3}$) $\times$ (3 $\times$ 10$^{\rm 2}$) =\\
%\vspace{0.3in}
%(8.0 $\times$ 10$^{\rm 18}$) $\div$ (4.0 $\times$ 10$^{\rm 14}$) =  \\

%\end{flushleft}
%Note how we rewrite the exponent to handle cases where the number out front
%is greater than 10, or less than 1.

\subsection{Calculator Issues}

Since you will be using calculators in nearly all of the labs this semester, you should become familiar with how to use them for functions beyond simple arithmetic.

\subsubsection{Scientific Notation on a Calculator}

Scientific notation on a calculator is usually designated with an ``E.''  For example, if you see the number 8.778046E11 on your calculator, this is the same as the number 8.778046 $\times 10^{11}$.  Similarly, 1.4672E-05 is equivalent to 1.4672 $\times 10^{-5}$.\\

\noindent Entering numbers in scientific notation into your calculator depends on layout of your calculator; we cannot tell you which buttons to push without seeing your specific calculator.  However, the ``E'' button described above is often used, so to enter 6.589 $\times 10^{7}$, you may need to type 6.589 ``E'' 7.\\

\noindent Verify that you can enter the following numbers into your calculator:\\
\begin{itemize}
\item  7.99921 $\times 10^{21}$
\item  2.2951324 $\times 10^{-6}$\\
\end{itemize}

%\noindent Performing arithmetic with numbers written in scientific notation works the same way as regular numbers.  Try these calculations:\\

\subsubsection{Order of Operations}

When performing a complex calculation, the order of operations is extremely important.  There are several rules that need to be followed:

\renewcommand{\labelenumi}{\roman{enumi}.}
\begin{enumerate}
\setcounter{enumi}{0}
\item Calculations must be done from left to right.
\item Calculations in brackets (parenthesis) are done first. When you have more than one set of brackets, do the inner brackets first.
\item Exponents (or radicals) must be done next.
\item Multiply and divide in the order the operations occur.
\item Add and subtract in the order the operations occur.
\end{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}

\noindent If you are using a calculator to enter a long equation, when in doubt as to whether the calculator will perform operations in the correct order, apply 
parentheses.\\

\noindent Use your calculator to perform the following calculations {\bf (2 points each):}\\

\begin{enumerate}
\setcounter{enumi}{19}
\item $\frac{(7 + 34)}{(2 + 23)}$ = \\
\item (4$^2$ + 5) $-$ 3 = \\
\item 20 $\div$ (12 $-$ 2) $\times$ 3$^2$ $-$ 2 = \\
\end{enumerate}

%\subsection{Algebra}

%Because this is a laboratory for a 100-level course, we do not use high-level mathematics.  However, we do sometimes encounter a little basic algebra and we need to briefly review the main concepts.  Algebra deals with equations and ``unknowns.''  Unknowns, or ``variables,'' are usually represented as a letter in an equation: $y$ = 3$x$ + 7. In this equation both ``$x$'' and ``$y$'' are variables. You do not know what the value of $y$ is until you assign a value to $x$. For example, if $x$ = 2, then $y$ = 13 ($y$ = 3$\times$2 + 7 = 13). Here are some additional examples:

%\begin{flushleft}
%y = 5x + 3, if x=1, what is y? Answer: y = 5$\times$1 + 3 = 5 + 3 = 8\\
%\vspace{0.3in}
%q = 3t + 9, if t=5, what is q? Answer: q = 3$\times$5 + 9 = 15 + 9 = 24\\
%\vspace{0.3in}
%y = 5x$^{\rm 2}$ + 3, if x=2, what is y? Answer: y = 5$\times$(2$^{\rm 2}$) + 3
%= 5$\times$4 + 3 = 20 + 3 = 23\\
%\vspace{0.3in}
%What is y if x = 6 in this equation: y = 3x + 13 =\\
%\end{flushleft}

%These problems were probably easy for you, but what happens when you have this  equation: y = 7x + 14, and you are asked to figure out what x is if y = 21?  Let's do this step by step, first we re-write the equation:
%\begin{flushleft}
%y = 7x + 14\\
%\vspace{0.2in}
%We now substitute the value of y (y = 21) into the equation:\\
%\vspace{0.2in}
%21 = 7x + 14\\
%\vspace{0.3in}
%Now, if we could get rid of that 14 we could solve this equation! Subtract 14 from both sides of the equation:\\
%\vspace{0.2in}
%21 $-$ 14 = 7x + 14 $-$ 14 \hspace{0.25in}(this gets rid of that pesky 14!)\\
%\vspace{0.2in}
%7 = 7x \hspace{0.25in}(divide both sides by 7)\\
%\vspace{0.2in}
%x = 1\\
%\end{flushleft}
%Ok, your turn: If you have the equation y = 4x + 16, and y = 8, what is x?
%\vspace{0.4in}

%We frequently encounter more complicated equations, such as y= 3x$^{\rm 2}$ + 2x $-$ 345, or p$^{\rm 2}$ = a$^{\rm 3}$. There are ways to solve such  equations, but that is beyond the scope of our introduction. However, you do need to be able to solve equations like this: y$^{\rm 2}$ = 3x + 3 (if you are told what ``x'' is!). Let's do this for x = 11:
%\begin{flushleft}
%Copy down the equation again:\\
%\vspace{0.2in}
%y$^{\rm 2}$ = 3x + 3 \\
%\vspace{0.2in}
%Substitute x = 11:\\
%\vspace{0.2in}
%y$^{\rm 2}$ = 3$\times$11 + 3 = 33 + 3 = 36\\
%\vspace{0.2in}
%Take the square root of both sides:\\
%\vspace{0.2in}
%(y$^{\rm 2}$)$^{\rm 1/2}$ = (36)$^{\rm 1/2}$\\
%\vspace{0.2in}
%y = 6
%\end{flushleft} 

%Did that make sense? To get rid of the square of a variable you have to take the square root: (y$^{\rm 2}$)$^{\rm 1/2}$ = y. So to solve for y$^{\rm 2}$, we took the square root of both sides of the equation.

\subsection{Graphing and/or Plotting}

Now we want to discuss graphing data. You probably learned about making graphs in high school. Astronomers frequently use graphs to plot data. You have probably seen all sorts of graphs, such as the plot of the performance of the stock market shown in Fig.~\ref{stock-fig}. A plot like this shows the history of the stock market versus time. The ``x'' (horizontal) axis represents time, and the ``y'' (vertical) axis represents the value of the stock market. Each place on the curve that shows the performance of the stock market is represented by two numbers, the date (x axis), and the value of the index (y axis).  For example, on May 10 of 2004, the Dow Jones index stood at 10,000.\\

\begin{figure}[htb]
\centerline{{\includegraphics[width=12cm]{stockplot.jpg}}}
\caption{The change in the Dow Jones stock index over one year (from April 
2003 to July 2004).}
\label{stock-fig}
\end{figure}

\noindent Plots like this require two data points to represent each point on the curve or in the plot. For comparing the stock market you need to plot the
value of the stocks versus the date. We call data of this type an ``ordered 
pair.'' Each data point requires a value for $x$ (the date) and $y$
(the value of the Dow Jones index).\\

\begin{table}[htb]
\begin{center}
\caption{Temperature vs. Altitude}
\begin{tabular}{|c|c|}
\hline
Altitude & Temperature\\
(feet) & $^{\rm o}$F\\
\hline
0       &  59.0\\
2,000   &  51.9\\
4,000   &  44.7\\
6,000   &  37.6\\
8,000   &  30.5\\
10,000  &  23.3\\
12,000  &  16.2\\
14,000  &  9.1\\
16,000  &  1.9\\
\hline
\end{tabular}
\label{mtntemps-tab}
\end{center}
\end{table}

\noindent Table~\ref{mtntemps-tab} contains data showing how the temperature changes with altitude near the Earth's surface. As you climb in altitude, the temperature goes down (this is why high mountains can have snow on them year round, even though they are located in warm areas). The data points in this table are plotted in Figure~\ref{temps-fig}.\\

\begin{figure}[htb]
\centerline{{\includegraphics[width=13cm]{tempvsalt.pdf}}}
\caption{The change in temperature as you climb in altitude with the
data from Table~\ref{mtntemps-tab}. At sea level (0 ft altitude) the surface
temperature is 59$^{\rm o}$F. As you go higher
in altitude, the temperature goes down.}
\label{temps-fig}
\end{figure}

%\subsubsection{Fitting a Line}

%\noindent Looking at the plot of temperature versus altitude, we see that a straight line can be drawn through the data points. We can figure out the equation of this straight line and then predict the temperature at any altitude. In high school you learned that the equation of a line was y = $m$x + b, where ``$m$'' is the ``slope'' of the line, and ``b'' is the ``y intercept.'' The y intercept is simply where the line crosses the y-axis. In the plot, the y intercept is at 59.0, so b = 59. So, we can rewrite the equation for this line as y = $m$x + 59.0.  How can we figure out $m$?  We pick {\it any other data point} and solve the equation--let's choose the data point at 10,000 feet. The temperature (y) is 23.3 at 10,000 feet (= x): 23.3 = 10000m + 59. Subtracting 59 from both sides shows 23.3 $-$ 59 = 10000x + 59 $-$ 59, or $-$35.7 = 10000m. To find $m$ we simply divide both sides by 10,000: m = $-$35.7/10000 = $-$0.00357. In scientific notation, the equation for the temperature vs. altitude is y = $-$3.57$\times$10$^{\rm -3}$x + 59.0. Why is the slope negative? What is happening here? As you go up in altitude, the temperature goes down.  Increasing the altitude (x) decreases the temperature (y). Thus, the slope has to be negative.

%\begin{enumerate}
%\setcounter{enumi}{22}
%\item  Using the equation for temperature versus altitude that you just derived, what is the temperature at 20,000 feet?  ({\bf 4 points})\\ \vspace{0.2in}
%\end{enumerate}

\subsubsection{The Mechanics of Plotting}

When you are asked to plot some data, there are several things to keep in mind.\\

\noindent  First of all, the plot axes {\bf must be labeled.}  This will be emphasized throughout the semester.  In order to quickly look at a graph and determine what information is being conveyed, it is imperative that both the x-axis and y-axis have labels.\\

\noindent  Secondly, if you are creating a plot, choose the numerical range for your axes such that the data fit nicely on the plot.  For example, if you were to plot the data shown in Table~\ref{mtntemps-tab}, with altitude on the y-axis, you would want to choose your range of y-values to be something like 0 to 18,000.  If, for example, you drew your y-axis going from 0 to 100,000, then all of the data would be compressed towards the lower portion of the page.  It is important to choose your {\it ranges} for the x and y axes so they bracket the data points.

\subsubsection{Plotting and Interpreting a Graph}

Table~\ref{twotemps-tab} contains hourly temperature data on January 19, 2006, for two locations: Tucson and Honolulu.
\begin{table}[htb]
\begin{center}
\caption{Hourly Temperature Data from 19 January 2006}
\begin{tabular}{|c|c|c|}
\hline
Time & Tucson Temp. & Honolulu Temp.\\
hh:mm & $^{\rm o}$F & $^{\rm o}$F\\
\hline
00:00 & 49.6 & 71.1\\
01:00 & 47.8 & 71.1\\
02:00 & 46.6 & 71.1\\
03:00 & 45.9 & 70.0\\
04:00 & 45.5 & 72.0\\
05:00 & 45.1 & 72.0\\
06:00 & 46.0 & 73.0\\
07:00 & 45.3 & 73.0\\
08:00 & 45.7 & 75.0\\
09:00 & 46.6 & 78.1\\
10:00 & 51.3 & 79.0\\
11:00 & 56.5 & 80.1\\
12:00 & 59.0 & 81.0\\
13:00 & 60.8 & 82.0\\
14:00 & 60.6 & 81.0\\
15:00 & 61.7 & 79.0\\
16:00 & 61.7 & 77.0\\
17:00 & 61.0 & 75.0\\
18:00 & 59.2 & 73.0\\
19:00 & 55.0 & 73.0\\
20:00 & 53.4 & 72.0\\
21:00 & 51.6 & 71.1\\
22:00 & 49.8 & 72.0\\
23:00 & 48.9 & 72.0\\
24:00 & 47.7 & 72.0\\
\hline
\end{tabular}
\label{twotemps-tab}
\end{center}
\end{table}

\begin{enumerate}
\setcounter{enumi}{22}
\item  On the blank sheet of graph paper in Figure~\ref{graphpaper-fig}, plot the hourly temperatures measured for Tucson and Honolulu on 19 January 2006.  ({\bf 10 points})\\
\item  Which city had the highest temperature on 19 January 2006? ({\bf 2 points})\\
\item  Which city had the highest {\it average} temperature? ({\bf 2 points})\\
\item  Which city heated up the fastest in the morning hours? ({\bf 2 points})\\
\end{enumerate}

\begin{figure}[htb]
\centerline{{\includegraphics[width=16cm]{graphpaper.pdf}}}
\caption{Graph paper for plotting the hourly temperatures in Tucson and Honolulu.}
\label{graphpaper-fig}
\end{figure}

\noindent While straight lines and perfect data show up in science from time to time, it is actually quite rare for $real$ data to fit perfectly on top of a line. One reason for this is that all measurements have {\it error}. So even though there might be a perfect relationship between x and y, the uncertainty of the measurements introduces small deviations from the line. In other cases,
the data are $approximated$ by a line. This is sometimes called a {\it best-fit}
relationship for the data.
%An example of a plot with real data is shown in Figure \ref{fig:mvporb}. In this case, the data suggest that there is a general trend between the absolute magnitude (M$_{\rm V}$) and the Orbital Period in certain types of binary stars. But some other factor plays a role in determining the final relationship, so some stars do not fit very well, and hence their absolute magnitudes cannot be estimated very well from their orbital periods (the vertical bars associated with each data point are error bars, and represent the measurement error).
%\begin{figure}[!ht]
%%\centering
%\leavevmode
%\epsfysize=0.7\textheight
%%\epsfbox{COMET/Asteroid460.eps}
%\epsfbox{LAB0/mvporbforlab.eps}
%\caption{The relationship between absolute visual magnitude (M$_{\rm V}$) and Orbital Period for catclysmic variable binary stars.}
%\label{fig:mvporb}
%\end{figure}

\subsection{Does it Make Sense?}

This is a question that you should be asking yourself after {\it every} calculation that you do in this class!\\

\noindent One of our primary goals this semester is to help you develop intuition about our solar system.  This includes recognizing if an answer that you get ``makes sense.''  For example, you may be told (or you may eventually know) that Mars is 1.5 AU from Earth.  You also know that the Moon is a lot closer to the Earth than Mars is.  So if you are asked to calculate the Earth-Moon distance and you get an answer of 4.5 AU, this should alarm you!  That would imply that the Moon is {\bf three times} farther away from Earth than Mars is!  And you know that's not right.\\

\noindent Use your intuition to answer the following questions.  In addition to just giving your answer, state {\it why} you gave the answer you did. ({\bf 5 points each})\\
\vspace{0.5in}\\
\begin{enumerate}
\setcounter{enumi}{26}
\item  Earth's diameter is 12,756 km.  Jupiter's diameter is about 11 times this amount.  Which makes more sense: Jupiter's diameter being 19,084 km or 139,822 km?\\
\vspace{0.75in}
\item  Sound travels through air at roughly 0.331 kilometers per second.  If BX 102 suddenly exploded, which would make more sense for when people in Mesilla (almost 5 km away) would hear the blast?  About 14.5 seconds later, or about 6.2 minutes later?\\
\vspace{0.75in}
\item  Water boils at 100 $^\circ$C.  Without knowing anything about the planet Pluto other than the fact that is roughly 40 times farther from the Sun than the Earth is, would you expect the surface temperature of Pluto to be closer to -100$^\circ$ or 50$^\circ$?\\
\vspace{0.75in}
\end{enumerate}

\noindent 

\subsection{Putting it All Together}

We have covered a lot of tools that you will need to become familiar with in order to complete the labs this semester.  Now let's see how these concepts can be used to answer real questions about our solar system. {\it Remember, ask yourself {\bf does this make sense?} for each answer that you get!}

\begin{enumerate}
\setcounter{enumi}{29}
\item  To travel from Las Cruces to New York City by car, you would drive 3585 km.  What is this distance in AU? ({\bf 10 points})\\
\vspace{1.5in}
\item  The Earth is 4.5 billion years old.  The dinosaurs were killed 65 million years ago due to a giant impact by a comet or asteroid that hit the Earth.  If we were to compress the history of the Earth from 4.5 billion years into one 24-hour day, at what time would the dinosaurs have been killed?  ({\bf 10 points})\\
\vspace{1.5in}
\item  When it was launched towards Pluto, the New Horizons spacecraft was 
traveling at 
approximately 20 kilometers per second.  How long did it take to reach Jupiter, which is roughly 4 AU from Earth?  [Hint: see the definition of an AU in Section 1.3 of this lab.]  ({\bf 7 points})\\
\vspace{1in}
\end{enumerate}



\vspace{8in}
\mbox{ }
