Lecture 1

Lecture 1#

Lab 1#

  • RR Lyrae (measure distance with luminosity variations)

Magnitudes & Extinction#

The relationship between apparent magnitude and absolute magnitude.

\[ m_\lambda = M_\lambda + 5\log_{10}(d) - 5 \]

If the source goes through some material such as a dust cloud, extinction would happen which may reduce its magnitude and redden the spectra.

\[ m_\lambda = M_\lambda + 5\log_{10}(d) - 5 + A_\lambda \]

The extinction affects the radiation modeled by an exponential decay.

\[ F_\lambda = F_{\lambda,0} e^{-\tau_\lambda} \]
\[ \tau_\lambda = \int n\sigma_\lambda\ \mathrm{d}s \]

  • \(\tau\) : optical depth

  • \(n\) : number density

  • \(\sigma_\lambda\) : wavelength-dependent particle size

We solve for \(A_\lambda\) by the difference of the apparent magnitudes (initial then extincted),

\[ \Delta m = -2.5 \log_{10}\left(\frac{F_\lambda}{F_{\lambda,0}}\right) \approx 1.086\tau_\lambda \]
\[ A_\lambda \equiv \Delta m \]

The color excess or selective extinction is the amount of color change (traditionally reddening) due to the extinction which are given by,

\[ E(B-V) = (B-V) - (B-V)_0 \]

The total extinciton is then the extinciton measured in the V-band (\(A_V\)). This is a normalization value for \(R_V\),

\[ R_V = \frac{A_V}{E(B-V)} \]

RR Lyrae#

RR Lyrae is a variable cepheid that are hydrogen-burning low-mass dying stars.

It is observed that the luminosity and period are positively proportioned \(L \approx P\).

\[ \langle{M_V}\rangle = a\log P + b \]
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