29:61 General Astronomy
Fall 2005
Constants and Formulas

Just the facts, Ma'am

Atmospheric Pressure

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

$\bullet$ Change in atmospheric pressure associated with increase in altitude $\Delta z$

$$\Delta p = -\rho g \Delta z$$ (1)

where $\rho$ is the gas density (kilograms/m$^3$), $g$ acceleration due to gravity at Earth's surface (9.8 m/sec$^2$).

$\bullet$ Perfect gas law

$$p V = N k_B T$$ (2)

$$p = n k_B T$$ (3)

$V=$ 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.

$\bullet$ Differential form of change in atmospheric pressure with altitude

$$\frac{dp}{dz} = -\left( \frac{mg}{k_B T}\right) p$$ (4)

where $m$ mass of atom or molecule composing the atmosphere.

$\bullet$ Pressure as function of height in an isothermal atmosphere

$$p(z) = p_0 \exp(-z/Z_0)$$ (5)

$\bullet$ Isothermal pressure scale height

$$Z_0 = \frac{k_B T}{m g}$$ (6)

The Orbit of the Moon and Eclipses

$\bullet$ The altitude angle of a celestial object at transit is

$$Al = \bar{\lambda} + \delta$$ (7)

where $\delta$ is the declination of the object, and $\bar{\lambda}$ is the complement of the latitude.

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

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

$\bullet$ The maximum distance $l$ in back of an object in which an umbral eclipse of the Sun will be seen is

$$l = \frac{d}{2\tan [\theta/2]}$$ (8)

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).

Eclipse Cycles and Precession

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

$\bullet$ Condition for repetition of eclipses:

$$mP_{nod} = nP_{syn}$$ (9)

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:

$$\vec{L} = m \vec{r} \times \vec{v}$$ (10)

$\bullet$ Definition of vector angular velocity $\vec{\omega}$, such that $\vert\omega\vert = \frac{d \theta}{dt}$,

$$\vec{\omega} = \frac{1}{r^2} \vec{r} \times \vec{v}$$ (11)

Then,

$$\vec{L} = r^2m \vec{\omega} = I \vec{\omega}$$ (12)

With $I$ the moment of inertia.

$\bullet$ Moment of inertia for uniform sphere of mass $M$ and radius $R$:

$$I = \frac{2}{5} M R^2$$ (13)

$\bullet$ Definition of torque: causes change in angular momentum with time:

$$\vec{\tau} = \frac{d \vec{L}}{dt}$$ (14)

Vector Cross Products, Newton's Laws, the Gravitational Force

$\bullet$ The definition of a vector cross product: The magnitude of a cross product is defined as follows. If

$$\vec{C} = \vec{A} \times \vec{B}$$ (15)

then the direction of $\vec{A} \times \vec{B}$ is given by the right hand rule. The magnitude of $\vec{A} \times \vec{B}$ is given by

$$\vert C\vert = \vert A\vert \vert B\vert \sin \theta$$ (16)

where $\theta$ is the angle between the two vectors. Another way of expressing it as follows. Draw up an array

$$\left[ \begin{array}{ccc} \hat{e}_x & \hat{e}_y & \hat{e}_z \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array}\right]$$ (17)

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

$$\vec{C} = (A_y B_z - B_y A_z) \hat{e}_x + (A_z B_x - B_z A_x) \hat{e}_y + (A_x B_y - B_x A_y) \hat{e}_z$$ (18)

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

Newton's Laws of Motion

1. 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.
2. If a force acts on an object, its momentum changes according to
   

   $$\vec{F} = \frac{\Delta \vec{p}}{\Delta t} = \frac{d \vec{p}}{dt}$$ (19)

   
   
   If the mass of the object acted upon stays constant, this simplifies to
   

   $$\vec{F} = m\frac{\Delta \vec{v}}{\Delta t} = m\frac{d \vec{v}}{dt} = m\vec{a}$$ (20)

   
   
   where $\vec{a}$ is the acceleration.

3. 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''.

$\bullet$ 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

$$a = \frac{v^2}{r}$$ (21)

$\bullet$ 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

$$\vert F\vert = \frac{GMm}{r^2}$$ (22)

where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

$\bullet$ 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

$$v = \sqrt \frac{GM}{r}$$ (23)

Kepler's Laws, Radioactive Decay, Physics of Atmospheres

Orbits and Kepler's Laws

$\bullet$ Equations for an ellipse:
In Cartesian coordinates:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (24)

where $a$ is the semimajor axis, $b$ is the semiminor axis.

In polar coordinates $(r, \theta)$ ,

$$r = a(1-\epsilon \cos \theta)$$ (25)

where $a$ is the semimajor axis and $\epsilon$ is the eccentricity of the ellipse.

Eccentricity in terms of $a$ and $b$,

$$\epsilon = \sqrt (1-(b/a)^2)$$ (26)

$\bullet$ Kepler's 3rd Law:

$$a^3 = P^2$$ (27)

$a$ = semimajor axis of planetary orbit in astronomical units, $P$ is the orbital period in years.

Radioisotope Dating

$\bullet$ 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.

$\bullet$ Exponential decay law:

$$N(t) = N_0 e^{-\alpha t}$$ (28)

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

$$\alpha = \frac{0.693}{T_{1/2}}$$ (29)

$\bullet$ Equation for determining age of formation of rock from ratio of isotopes.

$$A \rightarrow B1 +C$$ (30)

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

$$\left(\frac{N_{B1}}{N_{B2}}\right) = \frac{1 - e^{-\alpha t}... ...rac{N_A}{N_{B2}}\right) +\left( \frac{N_{B10}}{N_{B2}} \right)$$ (31)

where $N_{B10}$ was the number of nuclei of isotope $B1$ when the rock formed.

Physical Characteristics of the Planets

$\bullet$ Definition of density $\rho$

$$\rho = \frac{M}{V}$$ (32)

where $M$ is mass and $V$ is volume. Units of density are kilograms/m$^3$. Typical densities of common substances and astronomical objects are:

• water: 1000 kg/m$^3$
• rock: 2900 - 3900 kg/m$^3$
• aluminum: 2700 kg/m$^3$
• brass: 8600 kg/m$^3$
• lead: 11300 kg/m$^3$

Physics of Planetary Atmospheres

$\bullet$ Escape speed from a planet

$$V_{esc} = \left( \frac{2 G M}{R} \right)^{\frac{1}{2}}$$ (33)

where $M$ is the mass of the planet, and $R$ is its radius.

$\bullet$ root-mean-square (rms) molecular speed in a gas

$$V_{rms} = \left( \frac{3 k_B T}{m} \right)^{\frac{1}{2}}$$ (34)

where $T$ is the temperature (K), and $m$ is mass of the molecule or atom in the gas.

$\bullet$ Definition of the distribution function for molecular speeds

$$dN = N(v) dv$$ (35)

is the differential number of molecules with speeds in the range $v \rightarrow v+dv$.

$$N_0 = \int_0^{\infty}N(v)dv$$ (36)

where $N_0$ is the total number of molecules/m$^3$.

$\bullet$ The Maxwellian distribution function

$$N(v) = \frac{2N_0}{\sqrt 2 \pi} \left( \frac{m}{k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}$$ (37)

This distribution describes the true distribution for gases in planetary atmospheres, as well as most other astronomical gases.

$\bullet$ Condition for retention of planetary atmosphere over geological timescales

$$V_{rms} \leq f V_{esc}$$ (38)

where $f$ is a number between 1/6 and 1/4.

Planetary Magnetism

The Urey Cycle

$\bullet$ The mineralogical chemical cycle responsible for controlling the level of carbon dioxide in the Earth's atmosphere.

$$MgSiO_3 + CO_2 \rightarrow MgCO_3 + SiO_2$$ (39)

The arrow can point to the right or the left, depending on the temperature.

Photoionization

$\bullet$ The ionization of molecules by ultraviolet light in the upper atmosphere of the Earth (and elsewhere).

$$O_2 + h \nu \rightarrow O_2^{+} + e$$ (40)

where $h \nu$ represents the energy present in a photon of light.

The Lorentz Force

$\bullet$ The force acting on a charged particle moving in electric $\vec{E}$ and magnetic $\vec{B}$ fields is given by .

$$\vec{F} = q \vec{E} + q \vec{v} \times \vec{B}$$ (41)

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 Tesla, those of the electric field are Volts/meter.

Electromagnetic Radiation

$\bullet$ Light corresponds to electromagnetic waves with wavelengths in the range $4.0 \times 10^{-7} \leq \lambda \leq 7.0 \times 10^{-7}$ meters.

The Solar Constant

$\bullet$ The solar constant = 1370 W/m$^2$. For other planets, it is inversely proportional to the square of the distance from the Sun.

The Stefan Boltzmann Law

$\bullet$ 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.

$$S = \sigma T^4 \mbox{ Watts/m}^2$$ (42)

where $\sigma = 5.67 \times 10^{-8}$ Watts/m$^2$/K$^4$, and $T$ is the temperature in degrees Kelvin.

The Equilibrium Temperature of a Planet

$\bullet$ The equilibrium temperature of a planet, ignoring the greenhouse effect of its atmosphere (which often is a major correction) is

$$T_{eq} = \left[ \frac{(1-A)S_0}{\sigma}\right]^{\frac{1}{4}}$$ (43)

where $S_0$ is the solar constant for that planet, and $A$ is the albedo.

Wien's Law

$\bullet$ The wavelength at which a blackbody radiator is brightest, $\lambda_{max}$ is given by

$$\lambda_{max} =\frac{hc}{5 k_B T}$$ (44)

where $h = 6.626 \times 10^{-34}$ is Planck's constant, $c$ is the speed of light, and $T$ is the temperature.

The Roche Limit

$\bullet$ 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

$$r_R = 2.44 \left( \frac{\rho_M}{\rho_m} \right)^{\frac{1}{3}} R$$ (45)

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.

Resonant Perturbations

$\bullet$ A periodic perturbation is said to be resonant with a periodic system when

$$nP_n = mP_p$$ (46)

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.