Questions:

• Why is aperture photometry preferred over PSF photometry?

1. Optics

1.1. Opacity and Extinction

• See iPad notes from May 26.

• Limb darkening is caused by effective temperature gradient plus optical depth effective. In particular, see the Wikipedia diagram in iPad notes. Out to the limb, the $𝑙=1$ (photons barely escape) point is farther from the center where the temperature falls off.

1.2. Adaptive Optics

• You measure the shape of wavefronts based on a reference star or laser to determine how the atmosphere instantaneously distrots light. In live time, you correct for this with a shape changing mirror.

1.3. Lambertian Radiator

• Lambertian radiator appears uniformly bright according to Evan Kim. What about limb darkening

2. Planetary Atmospheres

Reference: Seager book and Heng book.

Consider a planet tidally locked to its host star, so that the dayside and nightside are fixed.

The energy of the atmosphere is given by

${𝑃}_{\text{int}}$ is the power from the planet interior. ${𝑃}_{\text{inc}}$ is the power incident on the dayside. ${𝑃}_{\text{out}}$ is the thermal emission of the atmosphere.

The bond albedo ${𝐴}_{𝐵}$ represents the proportion of energy scattered off the dayside when illuminated by a uniform beam perpendicular to its cross section. [TODO] Reconcile this definition with the formal one below.

Usually ${𝑃}_{\text{int}}$ is negligible so we will ignore it from hereon. The primary source of it for Jupiter is the release of gravitiational energy, and for Earth it is radioactive decay.

2.1. Equilibrium Temperature

Let ${𝐹}_{𝑠}$ and ${𝐹}_{𝑝}$ be the surface brightness (power per unit area) of the star and planet resp. At equilibrium, $\frac{\mathrm{d}𝑈}{\mathrm{d}𝑡}=0$, and we have

where $𝑓$ is a correction factor. Let ${𝑓}^{\prime }=𝑓/4$. For example, if there is no circulation between the dayside and nightside, we have ${𝑓}^{\prime }=2/4$. “If the atmosphere instantaneously reradiates the absorbed radiation (with no advection), ${𝑓}^{\prime }=\frac{2}{3}.$”

The equilbrium temperature is obtained by treating the planet as a blackbody ($𝐹=𝜎{𝑇}^4$):

Practically, we only make measurements in a specified bandpass $\left[{𝜈}_1,{𝜈}_2\right]$. The planetary brightness is the effective temperature of the planet wrt. this passband, i.e. the temperature of a blackbody that outputs the same spectrum/flux in $\left[{𝜈}_1,{𝜈}_2\right]$.

2.2. The Four Albedos :angry:

We have already discussed Bond albedo above. In total there are four useful albedo quantities: the single-scattering, geometric, Bond, and spherical albedo. We indiciate wavelength dependence with $𝜈$.

1. The single-scattering albedo $\widetilde{𝜔}\left(𝜈\right)$ is the fraction of incident light scattered by a given particle in the planetary atmosphere. For example in at visible wavelengths, $\widetilde{𝜔}<0.8$ for a water ice crystal in Earth’s atmosphere, $\widetilde{𝜔}\sim 0.1$ for a plant leaf, $0.35<\widetilde{𝜔}<0.8$ for clouds, $\widetilde{𝜔}<0.05$ for ocean water.
   • Particles such as ice crystals have a tendency to backscatter incident radiation in the opposite direction, akin to reflection. This nonisotropy is described in the particle’s bidirectional reflection distribution function.
2. The geometric albedo ${𝐴}_{𝑔}\left(𝜈\right)$ is the ratio of a planet’s flux at zero phase angle (star-planet-observer angle) to the flux from a Lambert disk at the same distance and cross sectional area.

   • Working definition: Define ${𝐴}_{𝑔}\left(𝜈\right)$ according to the secondary eclipse depth

     $$𝐹/{𝐹}_{\star }={𝐴}_{𝑔}{\left(\frac{{𝑅}_{𝑝}}{𝑎}\right)}^2.$$ (4)

   • Alternative definition: At zero phase angle, the amount of light reflected by the planet is

     $${𝐹}_{𝑝}=\frac{{𝐴}_{𝑠}\left(𝜈\right)\cdot {𝐹}_{\star }\left(𝜈\right){\left(\frac{{𝑅}_{\star }}{𝑎}\right)}^2𝜋{𝑅}_{𝑝}^2}{\frac{4𝜋}{𝑔}{𝑅}_{𝑝}^2}$$ (5)
     where $𝑔≔\frac{4{𝐴}_{𝑔}\left(𝜈\right)}{{𝐴}_{𝑠}\left(𝜈\right)}$ is a correction factor for anisotropic scattering. See Seager pg 43.
3. The bond albedo ${𝐴}_{𝐵}$ is the fraction of incident stellar energy scattered back into space by the planet integrated over all wavelengths and directions ([TODO] this is vague).
   • As Equation 6 shows, ${𝐴}_{𝐵}$ depends on the spectrum of the star.

4. The spherical albedo ${𝐴}_{𝑆}\left(𝜈\right)$ is analogous to the Bond albedo but at a specific frequency:

   $${𝐴}_{𝐵}=\frac{{\int }_0^{\infty }{𝐴}_{𝑆}\left(𝜈\right){𝐹}_{\text{inc}}\left(𝜈\right)\mathrm{d}𝜈}{{\int }_0^{\infty }{𝐹}_{\text{inc}}\left(𝜈\right)\mathrm{d}𝜈}.$$ (6)

These definitions are vague and circular. I’m content with the working definition of geometric albedo right now.

3. Cosmology

• https://web.mit.edu/8.286/www/lecn18/ln03-euf18.pdf
• Universe is currently matter dominated
• Newtonian cosmology: 2nd order IVP
  1. Initial positions: Homogeneous sphere of some radius ${𝑅}_{\text{max}}$ and density ${𝜌}_{𝑖}$
  2. Initial velocities: Hubble’s law: ${𝐯}_{𝑖}=𝐻\left(𝑡=0\right){𝐫}_{𝑖}$
  3. Time evolution ODE: Newton’s law
  • Conservation of energy gives Friedmann equation
  • The result is that everything expands by the same scale factor, independent of ${𝑅}_{\text{max}}$.
  • By convention, we say that $𝑎\left({𝑡}_0=13.7\text{Byr}\right)=1$, so scale factors in the past were less than 1.

4. USAAAO Cheatsheet

Observations

• Parallax: $𝑑=1/𝑝$ where $\left[𝑑\right]=\text{pc}$ and $\left[𝑝\right]=\text{arcsec}$
• Apparent magnitude: $𝑚-{𝑚}_0=-2.5{log}_{10}\left(𝐹/{𝐹}_0\right)$, where the reference ${𝑚}_0$ = 0 is Vega.
  • ${𝑚}_{\text{dot.circle}}=-26$
  • Absolute magnitude: $𝑀-𝑚=-2.5log\left({𝐹}_{\text{10pc}}/𝐹\right)=-2.5log\left({\left(10\text{pc}\right)}^2/{𝑑}^2\right)$

• Telescopes ($𝑓$ = objective lens focal length, $𝐷$ = aperture diameter)

  • $𝑓$ number: $\frac{𝑓}{𝐷}$
  • Rayleigh’s (somewhat arbitrary) criterion: $sin𝜃=1.22𝜆/𝐷$
  • Telescope magnification: $𝑚={𝑓}_{\text{obj}}/{𝑓}_{\text{eye}}$ where
  $$\frac{1}{𝑖}+\frac{1}{𝑜}=\frac{1}{𝑓}=\left(𝑛-1\right)\left(\frac{1}{{𝑅}_1}-\frac{1}{{𝑅}_2}\right)$$ (7)

Empirical relations

• Mass-luminosity relation for MS stars: $𝐿\propto {𝑀}^{3.5}$
• Wien’s law: ${𝜆}_{\text{peak}}𝑇=2.9\times {10}^{-3}\text{m}\cdot \text{K}$
• Hubble’s law: $𝑣={𝐻}_0𝑑$
  • Redshift: $𝑧=\mathrm{\Delta }𝜆/𝜆=\left({𝜆}_{\text{obs}}-{𝜆}_{\text{rest}}\right)/{𝜆}_{\text{rest}}\approx 𝛽$

Dynamics

• Virial theorem: $⟨\text{KE}⟩=-⟨𝑈⟩/2$, or more generally,

Spherical trigonometry

• Yikes

Misc

• Know the constellations
• Know different coordinate system like RA and DEC
• Remember hydrogen spectrum
  • Transitions to the ground state (Lyman series): emit ultraviolet photons
  • Transitions to the $𝑛=2$ state (Balmer series): emit visible photons
  • Transitions to the $𝑛=3$ state (Paschen series): emit infrared photos