Phys 590 - Final Exam

- Radar (in-solar-system)
- Stellar Parallax (out-solar-system)
- Distance Modulus
- Standard Candles (galaxies)

A parsec is the distance at which a star has a parallax angle $\pi^"$ equal to one arcsecond.

Stellar Parallax is the apparent motion of stars due to the Earth's motion around the Sun.

$d = \frac{206,265 \space arcsec}{\pi^"}$

Distance Modulus is the difference between a star's apparent and absolute magnitude.

$m - M = 5logd - 5$
$m - M = 5log(\frac{d}{10pc})$

The intrinsic brightness of a star is also known as it's luminosity. Luminosity is the rate at which a star emits energy in the form of elctromagnetic radiation.

Apparent brightness is the flux of a star. Flux is the rate per unit are at which a stars energy strikes a surface held perpendicula to a star.

$F = \frac{L}{4\pi d^2}$

The apparent magnitude is a measure of brightness of a star or another astronomical object. The more negative the brighter the object.

$m = C - 2.5logF$

Absolute magnitude is the apparent magnitude of a star if it were at a distance of 10 parsecs.

$M = m - 5logd + 5$

Bolometric flux is the total flux of a star integrated over all wavelengths.

$M_{bol} = 4.74 - 2.5log(\frac{L}{L_{sun}})$
$\frac{L}{L_{sun}} = 10^{0.4(4.74 - M_{bol})}$

Color index is the differnece of its apparent magnitude seen through two different filters. Using B-V, it allows you to determine the temperature of a star. If B-V > 0 = T > 10,000K and will be bluer than Vega. If B-V < 0 = T < 10,000K and will be redder than Vega.

$BC = M_{bol} - M_V$

If this value is a large negative value, it is either a cooler star than the Sun or a hotter star than the Sun. The closer the value is to 0, the closer type of star it is to the Sun.

$m - M = 5logd - 5 + A_V$

Correcting for extinction gives you a better distance measurement and also makes the star bluer due to removing interstellar reddening.

Using,

$R = \frac{A_V}{E(B-V)}$

where R usual equates to a value of 3.1.

$\frac{F_1}{F_2} = 100^{\frac{m_2 - m_1}{5}} = 2.512^{m_2 - m_1}$
or
$m_a - m_b = -2.5log(\frac{F_b}{F_a})$

It implied that a difference of 5 magnitudes equals a 100x difference in brightness/flux.

$\frac{\Delta{\lambda}}{\lambda} = \frac{\Delta{v}}{c}$

Using Kepler's 3rd Law,

$M_A + M_B = \frac{{4{\pi^2}{a^3}}}{GP^2}$

* Visual Binary - two stars of the binary are indivdually resolved with large separations and periods.
* Spectroscopic Binary - two stars are unresolved but the system shows absorption lines that oscillate as the radial velocity of the stars changes. They have small orbits and small periods.
* Eclipsing Binary - two stars are unresolved but the orbital plane is edge on in which we can detect a dip in flux. These binaries have small separations and short periods.

Visual binaries use a reduced form of Kepler's 3rd Law = $\frac{a^3}{P^2}$.

Spectroscopic binaries use the full form with $(a_A + a_B)^3$ which are calculated using radial velocities $v_a sini$ or $v_b sini$ in the period formula.

$m_1 r_1 = m_2 r_2$
$m_1 v_1 = m_2 v_2$

$\mu = \frac{\bar{m}}{m_H}$

Ionized:
$\mu (ionized) = \frac{\rho}{nm_p} = {(2X + 3/4Y + 1/2Z)}^{-1}$

Neutral:
$\mu (neutral) = {(X + \frac{Y}{4} + \frac{Z}{A})}^{-1}$

Hydrostatic Equilibrium is when the self gravity and internal pressure are equal.

$\frac{dP}{dr} = -\frac{GM_rp}{r^2}$

$P = nkT = \frac{\rho kT}{\mu m_p}$

Hot to Cold: QBAFGKMLT

Where including subdivisions, 1 is the hottest and 9 is the coolest.

O - 40000K
B - 20000K
A - 9000K
F - 7000K
G - 5500K
K - 4500K
M - 3000K
L - 2000K
T - <1300K

$H = \frac{kT}{g\mu m_p}$

Distance over which a physical quantity decreases by a factor of e (falling off).

Use,

${m_V - M_V} = 5logd - 5 + A_V$

$\frac{dM}{dr} = 4\pi r^2 \rho (r)$

This formula tells us that the total mass of a spherical star is the sum of masses of infinitesimally thin spherical shells.

$P(r) = \frac{p(r)kT(r)}{\mu m_p}$

This equation tells you the relationship among density, temperature, and pressure for a gas.

$T_c = \frac{M \mu}{R}$ for masses greater than
$1.66M_{sol}$.

$T_c = \frac{M}{R}$ for low mass stars.

$\frac{dT}{dr} = -\frac{3p(r)k(r)L(r)}{16\pi ac{T(r)}^3 r^2}$

or

$\frac{dT}{dr} = -\frac{3p(r)k(r)L(r)}{64\pi \sigma _{SB} {T(r)}^3 r^2}$

$\frac{dT}{dr} = (1-\frac{1}{\gamma})\frac{T(r)}{P(r)} \frac{dP}{dr}$

For simple atomic and fully ionized gases,
$\gamma = \frac{5}{3}$

For diatomic gases,
$\gamma = \frac{7}{5}$

$dL = (4\pi r^2 dr)\rho \epsilon$

or

$\frac{dL}{dr} = 4\pi r^2 \rho (r) \epsilon (r)$

Test particles moving from an initial point in a series of steps with the direction of each step being uncorrelated. The direction is independent of which its absorbed predecessor was traveling.

Fusion of 4 Hydrogen into Helium. ($T_c < 1.8 * 10^7 K$)

Two protons into H2, a positron, and neutrino.
$p + p => H^2 + e^+ + v_e$

Deutron fuses with protons into light Helium with excess energy carried away by gamma.
$H^2 + p => He^3 + \gamma$

Two light helium fuse to form normal Helium with excess protons spat out to repeat the chain.
$He^3 + He^3 => He^4 + p + p$

Fusion of 4 Hydrogen Atoms into Helium with Carbon, Nitrogen, and Oxygen as catalysts. ($T_c > 1.8 * 10^7 K$)

$C^{12} + p => N^{13} + \gamma$
$N^{13} => N^{13} + e^+ + v_e$
$C^{13} + p => N^{14} + \gamma$
$N^{14} + p => O^{15} + \gamma$
$O^{15} => N^{15} + e^+ + v_e$
$N^{15} + p => C^{12} + He^4$

Fusion of 3 Normal Helium into Carbon with beryllium as a catalyst.

$He^4 + He^4 => Be^8 + \gamma$
$He^4 + Be^8 => C^{12} + \gamma$

$mfp = \frac{1}{\sigma n}$

where sigma = cross-sectional area and n = number density.

Color excess is the observed color index subtracted by the intrinsic color index if there were no reddening.

- Cold Molecular Clouds
- Cold Neutral Medium (HI Regions)
- Warm Neutral Medium
- Warm Ionized Medium (HII Regions)
- Hot Ionized Medium

$T = 10K, n > 10^9 m^{-3}$

$T = 100K, n = 10^8 m^{-3}$

$T = 7000K, n = 4 * 10^5 m^{-3}$

$T = 10^4 K, n = 10^6 m^{-3}$

$T = 10^6 K, n < 10^4 m^{-3}$

When an electron is accelerated by a positively charged ion and produces a photon.

- 21 cm (Cool Neutral Medium)
- Radio Lines (GMC, Cool Neutral Medium)
- Absorbtion Lines (ISM Clouds, Cool)
- Emission Lines (Planetary Nebula, Hot)
- Bremsstrahlung/Synchrotron (Supernova Remnants)

1. Molecular Cloud is perturbed by a shock wave and starts to collapse.
2. Collapse forms a rotationally supported disk with a protostar at the center.
3. Hot gas moves along magnetic field lines which form stellar wind and carries much angular momentum.
4. In the disk, dust gathers together to form planetesimals and planetesimals collide to form planets. Remaining gas and dust is blown away by the winds.
5. Central regions become hot and dense enough that hydrogen fusion ignites turning the protostar into a star.

1. ISM Collapses ("free fall")
2. Protostar Formed [50 Myr]
3. Pre-Main Sequence
4. Main Sequence Star [10 Gyr, H -> He (core)]
5. Red Giant Branch [1 Gyr, H -> He (shell)]
Helium Flash (electron degeneracy)
6. Horizontal Branch [100 Myr, H -> He (shell), He -> C (core)]
7. Asymptomatic Giant Branch [20 Myr, H -> He (OS), He -> C (IS)]
8. Planetary Nebula [50 kyr}
9. White Dwarf [infinity]

- Cepheid Stars
- RR Lyrae

The stars pulsate due the expansion and contraction of the outer layers in an attempt to reach equilibrium. Therefore they are considered unstable.

$M_V = -2.76log(P/10 \space days) - 4.16$

Smallest stars about 13x the mass of Jupiter with Deuterium fusion.

$R = 0.01R_{sun} (\frac{M}{0.7M_{sun}})^{-\frac{1}{3}}$

$M_{Ch} = 1.4M_{sun}$
The maximum possible mass for a white dwarf.

- Rapid Rotation (conserved angular momentum)
- Supported by Neutron Degeneracy Pressure
- Strong Magnetic Fields
- Small Radius
- High Mass for Size

$R_{ns} = 11km (\frac{M_{ns}}{1.4M_{sun}})^{-\frac{1}{3}}$

- Iron Core grows larger than Chandrasekhar mass. Collapses.
- Protons and Electrons combine to form neutrons.
- Neutron sphere is compressed beyond neutron degeneracy pressure and bounces back.
- Core Bounce sends a shock wave sending out material.
- Neutrinos sent out interact with gas and heat it up causing the shock wave to move again.
- Outer layers of the star are ejected at high speeds.

A type of supernova with strong absorption lines of iron. They are created by a binary system with a white dwarf and red giant feeding into the white dwarf until it collapses and ejects a lot of iron into the interstellar medium.

In a spiral galaxy, stars are formed in the spiral arms because the arms contain a lot of molecular gas.

$F_{pressure} = F_{gravity}$

$F_{pressure} = dPA$
$F_{gravity} = -\frac{GM(pAdr)}{r^2}$

Use the Hydrostatic Balance equation
and
$M = \frac{4\pi}{3} p r^3$

and integrate from Pc to 0 and 0 to R.

1) Start with central pressure.
2) Assume $p = M/R^3$ and $P = pT_c$
3) Use radiative transport and plug in assumptions for Luminosity.
4) $L=M^\alpha$

Use $U = -q\frac{GM^2}{R}$
where q = 1.5.

$t_{kh} = \frac{U}{L}$

1) Remember $dF = 4\sigma_{sb} T^3 (dT/dr)dr$ and $dF = -k\rho Fdr$
2) Use $F = \frac{L}{4\pi r^2}$
3) Multiply by 4/3 and solve for dT/dr

1) Start with picture.
2) Remember $M = \frac{4\pi}{3}r^3 \rho$
and
$P^2 = \frac{4\pi^2 a^3}{GM}$.
3) P/2 = fft.

1) E = nkT, 2E = -U, U = -GMM/L, N=M/2m_h
2) Plug U into 2E = -U and N into E.
3) Equate using Virial.
4) Use $M = L^3 \rho$
5) Solve for L.

1) $\Delta{p} = \frac{\bar{h}}{\Delta{x}} = \bar{h} {n_{e}}^{\frac{1}{3}}$
2) $\Delta{v} = \frac{\Delta{p}}{m_e} = \bar{h} {n_{e}}^{\frac{1}{3}} / m_e$
3) $P = nkT = n_e m_e {v_e}^2$
4) Plug v into P.

1) Use $F_{rad} = \sigma_e F_p = \frac{\sigma_e L}{4\pi cr^2}$
2) Use $F_{grav} = -\frac{GM_{bh} m_p}{r^2}$
3) Equate and solve for L.

- Radio (strong in spirals)
- Infared (strong in spiral and irregular)
- Ultraviolet (star forming regions)
- X-rays (AGN or Neutron Star)

Use Virial Theorem
$2K = -U$
where
$U = -\frac{GM^2}{r}$
$K = \frac{1}{2}m<v^2>$
and where
$<v^2> = 3\sigma^2$
$\sigma = radial \space velocity \space dispersion$

Assuming the galaxy as a standard candle, the F-J relation is a relationship between $\sigma$ and $L_V$ for elliptical galaxies. For spiral galaxies in the T-F relation, peak rotation speed (v_c) and the absolute magnitude are related.

- Narrow/Broad Emission Lines
- UV Excess
- IR Excess
- AGN

- Lyman Limit Systems
- Metal Absorbers
- Lya Forest
- Damped Systems

Apparent motion that appears to be faster than the speed of light.

1) View the galaxy as a cube.
2) V = N/n
3) $l = V^{\frac{1}{3}}$
3) $P(N) = N*n*\sigma*l$

- Superclusters are not virialized and are filamentary or sheetlike.
- Voids cover 75% the volume of the universe and 15% of the mass.
- Fingers of god have a two-sided velocity effect <v^2> ~ 2sigma