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\begin{center}
{\bf 29:61 General Astronomy\\
     Fall 2005\\
     Constants and Formulas}\\

{\em Just the facts, Ma'am}
\end{center}

\begin{center}
\bf{Atmospheric Pressure}
\end{center}

\noindent
$\bullet$ Atmospheric pressure at sea level: $1.013 \times 10^5$ Newtons/m$^2$

\noindent 
$\bullet$ Change in atmospheric pressure associated with increase in altitude $\Delta z$
\begin{equation}
\Delta p = -\rho g \Delta z
\end{equation}
where $\rho$ is the gas density (kilograms/m$^3$), $g$ acceleration due to gravity at Earth's surface (9.8 m/sec$^2$).

\noindent
$\bullet$ Perfect gas law
\begin{eqnarray}
p V = N k_B T\\
p = n k_B T
\end{eqnarray}
$V=$ volume of gas, $N$ total number of particles (atoms and/or molecules) in the gas, $n$ number density of atoms and/or molecules (units are particles/m$^3$), $T$ is the temperature (degrees Kelvin), $k_B =1.3807 \times 10^{-23}$ J/K.  

\noindent
$\bullet$ Differential form of change in atmospheric pressure with altitude
\begin{equation}
\frac{dp}{dz} = -\left( \frac{mg}{k_B T}\right) p
\end{equation} 
where $m$ mass of atom or molecule composing the atmosphere.  

\noindent
$\bullet$ Pressure as function of height in an isothermal atmosphere
\begin{equation}
p(z) = p_0 \exp(-z/Z_0)
\end{equation}

\noindent
$\bullet$ Isothermal pressure scale height
\begin{equation}
Z_0 = \frac{k_B T}{m g}
\end{equation}


\begin{center}
{\bf  The Orbit of the Moon and Eclipses}\\
\end{center}

\noindent
$\bullet$ The altitude angle of a celestial object at transit is 
\begin{equation}
Al = \bar{\lambda} + \delta
\end{equation}
where $\delta$ is the declination of the object, and $\bar{\lambda}$ is the complement of the latitude.

\noindent
$\bullet$ The average inclination of the Moon's orbit to the plane of the ecliptic,
$\iota = 5^{\circ} 08^{'}$.  

\noindent
$\bullet$ The average angular diameter of the Sun as seen from Earth,
$\theta = 32^{'}$.  

\noindent
$\bullet$ The maximum distance $l$  in back of an object in which an umbral eclipse of the Sun will be seen  is 
\begin{equation}
l = \frac{d}{2\tan [\theta/2]}
\end{equation}
where $d$ is the diameter of the object, and $\theta$ is the angular diameter of the Sun at that point (32 arcminutes at the orbit of the Earth). 


\begin{center}
{\bf Eclipse Cycles and Precession}
\end{center}

\noindent
$\bullet$ Synodic period of the Moon: $P_{syn}=29.5306$ days.\\
$\bullet$ Nodal period of the Moon: $P_{nod}=27.2122$

\noindent
$\bullet$ Condition for repetition of eclipses:
\begin{equation}
mP_{nod} = nP_{syn}
\end{equation}
with $m$,$n$ integers

$\bullet$ Angular momentum of an object (with mass m) moving with velocity $\vec{v}$ a distance $\vec{r}$ from the origin of a coordinate system:
\begin{equation}
\vec{L} = m \vec{r} \times \vec{v}
\end{equation}

$\bullet$ Definition of vector angular velocity $\vec{\omega}$, such that $|\omega| = \frac{d \theta}{dt}$,
\begin{equation}
\vec{\omega} = \frac{1}{r^2} \vec{r} \times \vec{v}
\end{equation}
Then, 
\begin{equation}
\vec{L} = r^2m \vec{\omega} = I \vec{\omega}
\end{equation}
With $I$ the {\em moment of inertia}.  

$\bullet$ Moment of inertia for uniform sphere of mass $M$ and radius $R$:
\begin{equation}
I = \frac{2}{5} M R^2
\end{equation}

$\bullet$ Definition of torque: causes change in angular momentum with time:
\begin{equation}
\vec{\tau} = \frac{d \vec{L}}{dt}
\end{equation}


\begin{center}
{\bf Vector Cross Products, Newton's Laws, the Gravitational Force}\\
\end{center}

\noindent
$\bullet$ The definition of a vector cross product: 
The {\em magnitude} of a cross product is defined as follows.  If 
\begin{equation}
\vec{C} = \vec{A} \times \vec{B}
\end{equation}
then the {\em direction} of $\vec{A} \times \vec{B}$ is given by the right hand rule.  
The {\em magnitude} of $\vec{A} \times \vec{B}$ is given by 
\begin{equation}
|C| = |A| |B| \sin \theta
\end{equation}
where $\theta$ is the angle between the two vectors.  Another way of expressing it as follows.  Draw up an array 
\begin{equation}
\left[ 
\begin{array}{ccc}
\hat{e}_x & \hat{e}_y  & \hat{e}_z  \\
A_x  &  A_y  &  A_z  \\
B_x  &  B_y  &  B_z  
\end{array}
\right]
\end{equation}
For each component, knock out the column corresponding to that coordinate, and form the product of the remaining terms.  The result for the vector is
\begin{equation}
\vec{C} = (A_y B_z - B_y A_z) \hat{e}_x + (A_z B_x - B_z A_x) \hat{e}_y + (A_x B_y - B_x A_y) \hat{e}_z
\end{equation}

\noindent
$\bullet$ Remember that in a vector equation,  the equation must be satisfied {\em component by component}, in other words, you must satisfy the equation for the $x$ component of the vector, then the $y$ component, etc.  

\begin{center}
{\bf Newton's Laws of Motion}
\end{center}
\begin{enumerate}
\item An object in motion remains in motion with constant vector momentum $\vec{p}$, unless acted upon by an external force. An object at rest has zero momentum, and therefore remains at rest.  
\item If a force acts on an object, its momentum changes according to 
\begin{equation}
\vec{F} = \frac{\Delta \vec{p}}{\Delta t} = \frac{d \vec{p}}{dt}
\end{equation}
If the mass of the object acted upon stays constant, this simplifies to 
\begin{equation}
\vec{F} = m\frac{\Delta \vec{v}}{\Delta t} = m\frac{d \vec{v}}{dt} = m\vec{a}
\end{equation}
where $\vec{a}$ is the {\em acceleration}. 

\item If an object A exerts a force on B, B exerts a force on A which is equal in magnitude and opposite in direction to that exerted by A on B.  Rather lyrically said, ``to every action there is an opposite and equal reaction''.  
\end{enumerate}

\noindent
$\bullet$  Centripetal acceleration:  If an object moves in a circle of radius $r$ with speed $v$, it undergoes a centripetal acceleration which points toward the center of the circle and has a magnitude 
\begin{equation}
a = \frac{v^2}{r}
\end{equation}

\noindent
$\bullet$ The gravitational force: If two objects possessing masses $M$ and $m$ are a distance $r$ apart, there is an attractive force between them, the magnitude of which is 
\begin{equation}
|F| = \frac{GMm}{r^2}
\end{equation}  
where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

\noindent
$\bullet$ The circular orbit equation.  If $M \gg m$, and the orbit of $m$ about $M$ is circular, there is a relation between the radius of the orbit $r$, the orbital speed $v$, and the mass $M$ which is called the circular orbit equation.  It says 
\begin{equation}
v = \sqrt \frac{GM}{r}
\end{equation}

\begin{center}
{\bf Kepler's Laws, Radioactive Decay, Physics of Atmospheres}\\
\end{center}

\begin{center}
{\bf Orbits and Kepler's Laws}
\end{center}
\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. 

\begin{center}
{\bf Radioisotope Dating}
\end{center}

\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. 

\begin{center}
{\bf Physical Characteristics of the Planets}
\end{center}

\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}

\begin{center}
{\bf Physics of Planetary Atmospheres}
\end{center}

\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.  


\begin{center}
{\bf Planetary Magnetism}\\
\end{center}

\begin{center}
{\bf The Urey Cycle}
\end{center}
\noindent
$\bullet$ The mineralogical chemical cycle responsible for controlling the level of 
carbon dioxide in the Earth's atmosphere. 
\begin{equation}
MgSiO_3 + CO_2 \rightarrow MgCO_3 + SiO_2
\end{equation}
The arrow can point to the right or the left, depending on the temperature.  

\begin{center}
{\bf Photoionization}
\end{center}
\noindent
$\bullet$ The ionization of molecules by ultraviolet light in the upper atmosphere of the Earth (and elsewhere). 
\begin{equation}
O_2 + h \nu \rightarrow O_2^{+} + e
\end{equation}
where $h \nu$ represents the energy present in a photon of light.  

\begin{center}
{\bf The Lorentz Force}
\end{center}
\noindent
$\bullet$ The force acting on a charged particle moving in electric $\vec{E}$ and magnetic $\vec{B}$ fields is given by . 
\begin{equation}
\vec{F} = q \vec{E} + q \vec{v} \times \vec{B}
\end{equation}
where $q$ is the charge of the particle.  For a proton, the charge is $1.6022 \times 10^{-19}$ Coulombs, and for an electron, $q = -1.6022 \times 10^{-19}$ Often the symbol $e$ is used for this fundamental charge of an electron or proton.  The units of magnetic field are {\em Tesla}, those of the electric field are {\em Volts/meter}.  

\begin{center}
{\bf Electromagnetic Radiation}
\end{center}
\noindent
$\bullet$ Light corresponds to electromagnetic waves with wavelengths in the range 
$4.0 \times 10^{-7} \leq \lambda \leq 7.0 \times 10^{-7}$ meters.  

\begin{center}
{\bf The Solar Constant}
\end{center}
\noindent
$\bullet$ The solar constant = 1370 W/m$^2$.  For other planets,  it is inversely proportional to the square of the distance from the Sun. 

\begin{center}
{\bf The Stefan Boltzmann Law}
\end{center}
\noindent
$\bullet$ A perfect blackbody radiator radiates the following amount of power into space per unit of surface area.  The power consists of energy carried out by waves with a range of wavelengths.  
\begin{equation}
S = \sigma T^4 \mbox{  Watts/m}^2
\end{equation}
where $\sigma = 5.67 \times 10^{-8}$ Watts/m$^2$/K$^4$, and $T$ is the temperature in degrees Kelvin.  

\begin{center} 
{\bf The Equilibrium Temperature of a Planet}
\end{center}
$\bullet$ The equilibrium temperature of a planet, ignoring the greenhouse effect of its atmosphere (which often is a major correction) is
\begin{equation}
T_{eq} = \left[ \frac{(1-A)S_0}{\sigma}\right]^{\frac{1}{4}}
\end{equation}
where $S_0$ is the solar constant for that planet, and $A$ is the albedo.  

\begin{center} 
{\bf Wien's Law}
\end{center}
$\bullet$ The wavelength at which a blackbody radiator is brightest, $\lambda_{max}$ is given by
\begin{equation}
\lambda_{max} =\frac{hc}{5 k_B T}
\end{equation}
where $h = 6.626 \times 10^{-34}$ is Planck's constant, $c$ is the speed of light, and $T$ is the temperature.


\begin{center}
{\bf The Roche Limit}
\end{center}
\noindent
$\bullet$ The closest that a satellite can come to a planet (or small mass to a much larger mass) before it is torn apart by tidal stresses.  This means that the tides (differential ``stretching'' force across the object becomes larger than the gravitational force holding the object together.  The Roche limit is defined as a distance $r_R$, such that
\begin{equation}
r_R = 2.44 \left( \frac{\rho_M}{\rho_m} \right)^{\frac{1}{3}} R
\end{equation}
where $\rho_M$ is the density of the massive object, $\rho_m$ is the density of the smaller object (i.e. satellite), and $R$ is the radius of the massive object. 


\begin{center}
{\bf Resonant Perturbations}
\end{center}
\noindent
$\bullet$ A periodic perturbation is said to be {\em resonant} with a periodic system when 
\begin{equation}
nP_n = mP_p
\end{equation}
where $P_n$ is the natural period of the unperturbed system (think of the orbital period of a satellite around a planet), and $P_p$ is the period of the periodic perturbing force (think of the orbital period of a satellite further out) and $n$ and $m$ are any two integers.  When equation (2) is satisfied, the perturbation produces a large change in the orbital properties of the object being acted on.  For example, n=2, m=1 corresponds to the so-called 2:1 resonance, n=5, m=2 is the 5:2 resonance, etc.  All of these can be seen in the form of ``holes'' in the rings of Saturn and Kirkwood's Gaps in the asteroid belt.  






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