29:61 General Astronomy
Fall 2004
Lecture 10 ...October 21,2004
Kepler's Laws, Radioactive Decay, Physics of Atmospheres

Just the facts, Ma'am

Orbits and Kepler's Laws

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

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

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

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

$$r = a(1-\epsilon \cos \theta)$$ (2)

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

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

$$\epsilon = \sqrt (1-(b/a)^2)$$ (3)

$\bullet$ Kepler's 3rd Law:

$$a^3 = P^2$$ (4)

$a$ = 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:

$$N(t) = N_0 e^{-\alpha t}$$ (5)

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

$$\alpha = \frac{0.693}{T_{1/2}}$$ (6)

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

$$A \rightarrow B1 +C$$ (7)

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

$$\left(\frac{N_{B1}}{N_{B2}}\right) = \frac{1 - e^{-\alpha t}... ...rac{N_A}{N_{B2}}\right) +\left( \frac{N_{B10}}{N_{B2}} \right)$$ (8)

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

Physical Characteristics of the Planets

$\bullet$ Definition of density $\rho$

$$\rho = \frac{M}{V}$$ (9)

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

$$V_{esc} = \left( \frac{2 G M}{R} \right)^{\frac{1}{2}}$$ (10)

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

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

$$V_{rms} = \left( \frac{3 k_B T}{m} \right)^{\frac{1}{2}}$$ (11)

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

$$dN = N(v) dv$$ (12)

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

$$N_0 = \int_0^{\infty}N(v)dv$$ (13)

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

$\bullet$ The Maxwellian distribution function

$$N(v) = \frac{2N_0}{\sqrt 2 \pi} \left( \frac{m}{k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}$$ (14)

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

$$V_{rms} \leq f V_{esc}$$ (15)

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