Observational Astronomy
EQUATIONS
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INFORMATION
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$$ \large cos(\zeta) = sin(\phi)sin(\delta) + cos(\phi)cos(\delta)cos(h ) $$
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\(\\ {\zeta} = solar \hspace{4 pt} zenith \hspace{4 pt} angle \hspace{4 pt} (deg) \\ {\phi} = latitude \hspace{4 pt} (deg) \\ {\delta} = declination \hspace{4 pt} (deg) \\ \textbf{h} = hour \hspace{4 pt} angle \hspace{4 pt} (deg)\) |
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$$ \large
D = \frac{\alpha d}{206265} $$
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\(\\ \textbf{D} = diameter \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} object \hspace{4 pt} (km) \\ {\alpha} = angular \hspace{4 pt} size \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} (arcsec) \\ \textbf{d} = distance \hspace{4 pt} to \hspace{4 pt} object \hspace{4 pt} (km)\) |
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$$ \large
\Theta = \frac{D}{f_o} $$
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\({\Theta} = angular \hspace{4 pt} diameter \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} field \\ \hspace{8 pt} of \hspace{4 pt} view \hspace{4 pt} (m) \\ \textbf{D} = diameter \hspace{4 pt} of \hspace{4 pt} aperture \hspace{4 pt} in \hspace{4 pt} \\ \hspace{8 pt} eye-peice \hspace{4 pt} (m) \\ {f_o} = effective \hspace{4 pt} focal \hspace{4 pt} length \hspace{4 pt} (m)\) |
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$$ \large \alpha_c = \frac{1.22 \lambda}{D_o} $$
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\(\\ {\alpha_c} = Limit \hspace{4 pt} of \hspace{4 pt} Angular \hspace{4 pt} Resolution / \\ \hspace{10 pt} Rayleigh \hspace{4 pt} criterion \hspace{4 pt} (rad) \\ {\lambda} = average \hspace{4 pt} wavelength \hspace{4 pt} of \hspace{4 pt} light \\ \hspace{4 pt} contributing \hspace{4 pt} to \hspace{4 pt} image \hspace{4 pt} (m) \\ {D_o} = aperture \hspace{4 pt} (m)\) |
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$$ \large
\theta = 2.5 \hspace{4 pt} x \hspace{4 pt} 10^5 \hspace{4 pt} \frac{\lambda}{D} $$
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\(\\ {\theta} = diffraction-limited \hspace{4 pt} angular \\ \hspace{4 pt} resolution \hspace{4 pt} of \hspace{4 pt} a \hspace{4 pt} telescope \hspace{4 pt} (arcsec) \\ {\lambda} = wavelength \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (m) \\ \textbf{D} = diameter \hspace{4 pt} of \hspace{4 pt} telescope \hspace{4 pt} \\ \hspace{8 pt} objective \hspace{4 pt} (m)\) |
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$$ \large v = \frac{c}{n} $$
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\( \textbf{v} = velocity \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} in \\ \hspace{8 pt} medium \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \\ \hspace{8 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^{8} \hspace{2 pt} m \hspace{2 pt} s^{-1} \\ \textbf{n} = index \hspace{4 pt} of \hspace{4 pt} refraction \) |
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$$ \large m' - m = \varepsilon X - \zeta $$
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\( \textbf{m'} = instrumental \hspace{4 pt} magnitude \\ \textbf{m} = catalog \hspace{4 pt} magnitude \\ \epsilon = extinction \hspace{4 pt} coefficient \\ \textbf{X} = airmass \\ \zeta = zero - point \hspace{4 pt} offset \hspace{4 pt} between \\ \hspace{4 pt} instrumental \hspace{4 pt} and \hspace{4 pt} catalog \hspace{4 pt} magnitudes \) |
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$$ \large \sigma = \sqrt{\frac{x_i - \left \langle x \right \rangle}{n}} $$
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\(\\ {\sigma} = standard \hspace{4 pt} deviation \\ {x_i} = value \hspace{4 pt} of \hspace{4 pt} measurement \hspace{4 pt} i \\ {\left \langle x \right \rangle} = mean \hspace{4 pt} value \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} measured \hspace{4 pt} values \\\) |
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$$ \large \sigma_{n-1} = \sqrt{\frac{x_i - \left \langle x \right \rangle}{n-1}} $$
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\(\\ {\sigma_{n-1}} = standard \hspace{4 pt} deviation \\ {x_i} = value \hspace{4 pt} of \hspace{4 pt} measurement \hspace{4 pt} i \\ {\left \langle x \right \rangle} = mean \hspace{4 pt} value \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} measured \hspace{4 pt} values \\\) |
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$$ \large \sigma_m = \frac{\sigma}{\sqrt{n}} $$
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\(\\ {\sigma_m} = uncertainty \hspace{4 pt} in \hspace{4 pt} the \hspace{4 pt} mean \\ {\sigma} = population \hspace{4 pt} standard \hspace{4 pt} deviation \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} measured \hspace{4 pt} values\) |
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$$ \large {\chi_r}^2 = \frac{\Sigma(y_i - y(x_i))^2}{\sigma^2 \hspace{2 pt} (N-P)} $$
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\(\\ {\chi_r} = chi \hspace{4 pt} statistical \hspace{4 pt} metric \\ {y_i} = observed \hspace{4 pt} value \\ {y(x_i)} = expected \hspace{4 pt} value \\ {\sigma} = standard \hspace{4 pt} deviation \\ \textbf{N} = number \hspace{4 pt} of \hspace{4 pt} degrees \hspace{4 pt} of \\ \hspace{8 pt} freedom\\ \textbf{P} = number \hspace{4 pt} of \hspace{4 pt} free \hspace{4 pt} parameters\) |
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