User:Albert.white/notes

http://en.wikipedia.org/wiki/Wikipedia:Vandalism

$$ f= \frac{number ~ of ~ periods}{ duration ~of~ sample}$$

$$L=10^{\gamma} \times P^{\alpha} \times \dot{P}^{\beta}$$

$$S_{mean}= \frac{(S/N) T_{sys}}{G\sqrt{n_{p}t_{obs}\Delta{f}}}\sqrt{\frac{W}{P-W}}$$

$$ Phase (t) = f \times (t-t_{0}) - INT(f \times (t-t_{0})) $$

$$ n= \frac{v \ddot{v}}{\dot v ^2} \quad \quad ; \quad \quad v = \frac{1}{P} $$

$$ B= \left( \frac{3Ic^3P\dot{P}}{8\pi^{2}R^{6}} \right) $$

$$ B \approx 3.2 \times 10^{19} \sqrt{ P \dot{P}} ~  G $$

$$ t= -f \left[ (n-1) \frac{df}{dt}\right] ^{-1}   \left[ 1-   \left( \frac{f}{f_{0}} \right) ^{n-1} \right] $$

$$ t= -f \left( \frac {2df}{dt} \right) ^ {-1} = \frac{P}{2\ddot{P}} $$

$$\theta = \arcsin \left( \frac {\sqrt{B^{2}+2AB}} {A+B} \right)\quad\quad\quad\quad(1)$$

$$\frac {S}{N} = \frac {N_{*}}{\sqrt{N_{*} + n_{pix}(N_{S}+N_{D}+N_{R}^2)}}$$

$$\frac {S}{N} = \frac {N_{*}}{\sqrt{N_{*} + n_{pix}(1+\frac{n_{pix}}{n_{B}})(N_{S}+N_{D}+N_{R}^2 + G^2\sigma_{f}^2)}}$$

$$D = c \times (TR-TS)\quad\quad\quad\quad(2)$$

$$(1+x)^{1/2} = 1 + \frac{x}{2} + \frac{x^{2}}{8}... \quad\quad\quad\quad(11) $$

$$d\tau^{2} = \left( 1 - \frac{2M}{r}\right) dt^{2} - \frac{dr^{2}}{1- \frac{2M}{r}} - r^{2}d\Phi^{2}\quad\quad\quad\quad(3)$$

$$\left(\frac{d\tau}{dt}\right)^{2} = \left( 1 - \frac{2M}{r} \right) -r^{2}\left(\frac{d\Phi}{dt}\right)^{2} = \left(1-\frac{2M}{r}\right)-v^{2}\quad\quad\quad\quad(4)$$

$$\left( \frac{dt_{satllite}}{dt_{Earth}} \right)^{2} = \frac{1-\frac{2M}{r_{satellite}} -v^{2}_{satellite}} {1-\frac{2M}{r_{Earth}} -v^{2}_{Earth}}\quad\quad\quad\quad(5)$$

$$ \frac{dt_{satllite}}{dt_{Earth}} = \frac{\sqrt{ 1-\frac{2M}{r_{satellite}}}} {\sqrt{ 1-\frac{2M}{r_{Earth}}}}\quad\quad\quad\quad(9)$$

$$ \frac{dt_{satllite}}{dt_{Earth}} \approx 1 + \frac{M}{r_{satellite}} -\frac{v^{2}_{satellite}}{2} + \frac{M}{r_{Earth}} + \frac{v^{2}_{Earth}}{2}\quad\quad\quad\quad(8)$$

$$ \frac{dt_{satllite}}{dt_{Earth}} \approx 1 + \frac{M}{r_{Earth}} -\frac{M}{r_{satellite}} = 1+5.287 \times 10^{-10}\quad\quad\quad\quad(10)$$

$$M=\frac{GM}{c^{2}}\quad\quad\quad\quad(6)$$

$$v_{satellite} = 1.29 \times 10^{-5} ; v_{Earth} = 1.547 \times 10^{-6}\quad\quad\quad\quad(7)$$

$$x=-\frac{2M}{r_{satellite}} -v^{2}_{satellite}\quad\quad\quad\quad(12)$$

$$(1 -\frac{2M}{r_{sat}} -v^{2}_{sat})^{1/2} \approx 1+ \frac{\left(-\frac{2M}{r_{sat}} -v^{2}_{sat}\right)}{2} + \frac{\left(-\frac{2M}{r_{sat}} -v^{2}_{sat}\right)^{2}}{8}... (13)$$

$$(1 -\frac{2M}{r_{sat}} -v^{2}_{sat})^{1/2} \approx 1 - \frac{M}{r_{sat}} - \frac{v^{2}_{sat}}{2} \quad\quad\quad\quad (13b)$$

$$(1 -\frac{2M}{r_{Earth}} -v^{2}_{Earth})^{- 1/2} \approx 1 + \frac{M}{r_{Earth}} + \frac{v^{2}_{Earth}}{2} \quad\quad\quad\quad (13c)$$

$$\frac{dt_{satllite}}{dt_{Earth}} \approx \left(1 - \frac{M}{r_{sat}} - \frac{v^{2}_{sat}}{2}\right) \times \left(1 + \frac{M}{r_{Earth}} + \frac{v^{2}_{Earth}}{2}\right)$$

\approx

1+ \frac{M}{r_{Earth}} + \frac{v^{2}_{Earth}}{2} - \frac{M}{r_{sat}} - \frac{M^{2}}{r_{sat}r_{Earth}} - \frac{M v^{2}_{Earth}}{2r_{sat}} - \frac{v^{2}_{sat}}{2} - \frac{Mv^{2}_{sat}}{2r_{Earth}} - \frac{v^{2}_{sat}v^{2}_{Earth}}{4}

\quad\quad(14) $$

\approx

1+ \frac{M}{r_{Earth}} + \frac{v^{2}_{Earth}}{2} - \frac{M}{r_{sat}} - \frac{v^{2}_{sat}}{2}

\quad\quad(15) $$ $$= $$

$$$$

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