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{\bf 29:61 General Astronomy\\
     Fall 2004\\
     Lecture 10 \ldots October 21,2004\\
     Kepler's Laws, Radioactive Decay, Physics of Atmospheres}\\

{\em Just the facts, Ma'am}
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\begin{center}
{\bf Orbits and Kepler's Laws}
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\noindent
$\bullet$ Equations for an ellipse: \\
In Cartesian coordinates: 
\begin{equation}
 \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\end{equation}
where $a$ is the semimajor axis, $b$ is the semiminor axis.  

In polar coordinates $(r, \theta)$ , 
\begin{equation}
r = a(1-\epsilon \cos \theta)
\end{equation}
where $a$ is the semimajor axis and $\epsilon$ is the eccentricity of the ellipse. 

Eccentricity in terms of $a$ and $b$, 
\begin{equation}
\epsilon = \sqrt (1-(b/a)^2)
\end{equation}

\noindent
$\bullet$ Kepler's 3rd Law: 
\begin{equation}
a^3 = P^2
\end{equation}
$a$ = semimajor axis of planetary orbit in astronomical units, $P$ is the orbital period in years. 

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{\bf Radioisotope Dating}
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\noindent
$\bullet$ Radioactive decay, $A \rightarrow B+C$, where $A$ is the unstable parent isotope, $B$ is the daughter isotope (product of the decay), and $C$ is a particle which comes out as a result of the decay, such as beta particle (electron or positron), alpha particle (helium nucleus), or larger piece of a nucleus.  

\noindent
$\bullet$ Exponential decay law: 
\begin{equation}
N(t) = N_0 e^{-\alpha t}
\end{equation}
where $N_0$ is the number of parent nuclei at $t=0$, $\alpha$ is the decay constant, and $N(t)$ is the number of parent nuclei at time $t$.  The decay constant is related to the half life $T_{1/2}$ by 
\begin{equation}
\alpha = \frac{0.693}{T_{1/2}}
\end{equation}

\noindent
$\bullet$ Equation for determining age of formation of rock from ratio of isotopes. 
\begin{equation}
A \rightarrow B1 +C
\end{equation}
where $A$ is the radioactive parent isotope, $B1$ is the isotope of element $B$ that is the daughter product of the decay reaction, and $B2$ is the isotope of element $B$ that is not the daughter product of the decay.  Let $N_A$ be the number of isotopes of $A$ in a sample, $N_{B1}$ the number of isotopes of $B1$, and $N_{B2}$ the number of isotopes of $B2$, then we have the following equation
\begin{equation}
\left(\frac{N_{B1}}{N_{B2}}\right) = \frac{1 - e^{-\alpha t}}{e^{-\alpha t}} \left( \frac{N_A}{N_{B2}}\right) +\left( \frac{N_{B10}}{N_{B2}} \right)
\end{equation}
where $N_{B10}$ was the number of nuclei of isotope $B1$ when the rock formed. 

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{\bf Physical Characteristics of the Planets}
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\noindent
$\bullet$ Definition of density $\rho$ 
\begin{equation}
\rho = \frac{M}{V}
\end{equation}
where $M$ is mass and $V$ is volume.  Units of density are kilograms/m$^3$.  Typical densities of common substances and astronomical objects are: 
\begin{itemize}
\item water: 1000 kg/m$^3$
\item rock: 2900 - 3900  kg/m$^3$
\item aluminum: 2700 kg/m$^3$
\item brass: 8600 kg/m$^3$
\item lead: 11300 kg/m$^3$
\end{itemize}

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{\bf Physics of Planetary Atmospheres}
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\noindent
$\bullet$ Escape speed from a planet
\begin{equation}
V_{esc} = \left( \frac{2 G M}{R} \right)^{\frac{1}{2}}
\end{equation}
where $M$ is the mass of the planet, and $R$ is its radius. 

\noindent
$\bullet$ root-mean-square (rms) molecular speed in a gas
\begin{equation}
V_{rms} =  \left( \frac{3 k_B T}{m}  \right)^{\frac{1}{2}}
\end{equation}
where $T$ is the temperature (K), and $m$ is mass of the molecule or atom in the gas. 

\noindent
$\bullet$ Definition of the distribution function for molecular speeds 
\begin{equation}
dN = N(v) dv
\end{equation}
is the differential number of molecules with speeds in the range $v \rightarrow v+dv$.  
\begin{equation}
N_0 = \int_0^{\infty}N(v)dv
\end{equation}
where $N_0$ is the {\em total} number of molecules/m$^3$.  

\noindent
$\bullet$ The Maxwellian distribution function 
\begin{equation}
N(v) = \frac{2N_0}{\sqrt 2 \pi} \left( \frac{m}{k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}
\end{equation}
This distribution describes the true distribution for gases in planetary atmospheres, as well as most other astronomical gases. 

\noindent
$\bullet$ Condition for retention of planetary atmosphere over geological timescales
\begin{equation}
V_{rms} \leq f V_{esc}
\end{equation}
where $f$ is a number between 1/6 and 1/4.  


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