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f = n u m b e r o f p e r i o d s d u r a t i o n o f s a m p l e {\displaystyle f={\frac {number~of~periods}{duration~of~sample}}}
L = 10 γ × P α × P ˙ β {\displaystyle L=10^{\gamma }\times P^{\alpha }\times {\dot {P}}^{\beta }}
S m e a n = ( S / N ) T s y s G n p t o b s Δ f W P − W {\displaystyle S_{mean}={\frac {(S/N)T_{sys}}{G{\sqrt {n_{p}t_{obs}\Delta {f}}}}}{\sqrt {\frac {W}{P-W}}}}
P h a s e ( t ) = f × ( t − t 0 ) − I N T ( f × ( t − t 0 ) ) {\displaystyle Phase(t)=f\times (t-t_{0})-INT(f\times (t-t_{0}))}
n = v v ¨ v ˙ 2 ; v = 1 P {\displaystyle n={\frac {v{\ddot {v}}}{{\dot {v}}^{2}}}\quad \quad ;\quad \quad v={\frac {1}{P}}}
B = ( 3 I c 3 P P ˙ 8 π 2 R 6 ) {\displaystyle B=\left({\frac {3Ic^{3}P{\dot {P}}}{8\pi ^{2}R^{6}}}\right)}
B ≈ 3.2 × 10 19 P P ˙ G {\displaystyle B\approx 3.2\times 10^{19}{\sqrt {P{\dot {P}}}}~G}
t = − f [ ( n − 1 ) d f d t ] − 1 [ 1 − ( f f 0 ) n − 1 ] {\displaystyle t=-f\left[(n-1){\frac {df}{dt}}\right]^{-1}\left[1-\left({\frac {f}{f_{0}}}\right)^{n-1}\right]}
t = − f ( 2 d f d t ) − 1 = P 2 P ¨ {\displaystyle t=-f\left({\frac {2df}{dt}}\right)^{-1}={\frac {P}{2{\ddot {P}}}}}
θ = arcsin ( B 2 + 2 A B A + B ) ( 1 ) {\displaystyle \theta =\arcsin \left({\frac {\sqrt {B^{2}+2AB}}{A+B}}\right)\quad \quad \quad \quad (1)}
S N = N ∗ N ∗ + n p i x ( N S + N D + N R 2 ) {\displaystyle {\frac {S}{N}}={\frac {N_{*}}{\sqrt {N_{*}+n_{pix}(N_{S}+N_{D}+N_{R}^{2})}}}}
S N = N ∗ N ∗ + n p i x ( 1 + n p i x n B ) ( N S + N D + N R 2 + G 2 σ f 2 ) {\displaystyle {\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 × ( T R − T S ) ( 2 ) {\displaystyle D=c\times (TR-TS)\quad \quad \quad \quad (2)}
( 1 + x ) 1 / 2 = 1 + x 2 + x 2 8 . . . ( 11 ) {\displaystyle (1+x)^{1/2}=1+{\frac {x}{2}}+{\frac {x^{2}}{8}}...\quad \quad \quad \quad (11)}
d τ 2 = ( 1 − 2 M r ) d t 2 − d r 2 1 − 2 M r − r 2 d Φ 2 ( 3 ) {\displaystyle 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)}
( d τ d t ) 2 = ( 1 − 2 M r ) − r 2 ( d Φ d t ) 2 = ( 1 − 2 M r ) − v 2 ( 4 ) {\displaystyle \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)}
( d t s a t l l i t e d t E a r t h ) 2 = 1 − 2 M r s a t e l l i t e − v s a t e l l i t e 2 1 − 2 M r E a r t h − v E a r t h 2 ( 5 ) {\displaystyle \left({\frac {dt_{satllite}}{dt_{Earth}}}\right)^{2}={\frac {1-{\frac {2M}{r_{satellite}}}-v_{satellite}^{2}}{1-{\frac {2M}{r_{Earth}}}-v_{Earth}^{2}}}\quad \quad \quad \quad (5)}
d t s a t l l i t e d t E a r t h = 1 − 2 M r s a t e l l i t e 1 − 2 M r E a r t h ( 9 ) {\displaystyle {\frac {dt_{satllite}}{dt_{Earth}}}={\frac {\sqrt {1-{\frac {2M}{r_{satellite}}}}}{\sqrt {1-{\frac {2M}{r_{Earth}}}}}}\quad \quad \quad \quad (9)}
d t s a t l l i t e d t E a r t h ≈ 1 + M r s a t e l l i t e − v s a t e l l i t e 2 2 + M r E a r t h + v E a r t h 2 2 ( 8 ) {\displaystyle {\frac {dt_{satllite}}{dt_{Earth}}}\approx 1+{\frac {M}{r_{satellite}}}-{\frac {v_{satellite}^{2}}{2}}+{\frac {M}{r_{Earth}}}+{\frac {v_{Earth}^{2}}{2}}\quad \quad \quad \quad (8)}
d t s a t l l i t e d t E a r t h ≈ 1 + M r E a r t h − M r s a t e l l i t e = 1 + 5.287 × 10 − 10 ( 10 ) {\displaystyle {\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 = G M c 2 ( 6 ) {\displaystyle M={\frac {GM}{c^{2}}}\quad \quad \quad \quad (6)}
v s a t e l l i t e = 1.29 × 10 − 5 ; v E a r t h = 1.547 × 10 − 6 ( 7 ) {\displaystyle v_{satellite}=1.29\times 10^{-5};v_{Earth}=1.547\times 10^{-6}\quad \quad \quad \quad (7)}
x = − 2 M r s a t e l l i t e − v s a t e l l i t e 2 ( 12 ) {\displaystyle x=-{\frac {2M}{r_{satellite}}}-v_{satellite}^{2}\quad \quad \quad \quad (12)}
( 1 − 2 M r s a t − v s a t 2 ) 1 / 2 ≈ 1 + ( − 2 M r s a t − v s a t 2 ) 2 + ( − 2 M r s a t − v s a t 2 ) 2 8 . . . ( 13 ) {\displaystyle (1-{\frac {2M}{r_{sat}}}-v_{sat}^{2})^{1/2}\approx 1+{\frac {\left(-{\frac {2M}{r_{sat}}}-v_{sat}^{2}\right)}{2}}+{\frac {\left(-{\frac {2M}{r_{sat}}}-v_{sat}^{2}\right)^{2}}{8}}...(13)}
( 1 − 2 M r s a t − v s a t 2 ) 1 / 2 ≈ 1 − M r s a t − v s a t 2 2 ( 13 b ) {\displaystyle (1-{\frac {2M}{r_{sat}}}-v_{sat}^{2})^{1/2}\approx 1-{\frac {M}{r_{sat}}}-{\frac {v_{sat}^{2}}{2}}\quad \quad \quad \quad (13b)}
( 1 − 2 M r E a r t h − v E a r t h 2 ) − 1 / 2 ≈ 1 + M r E a r t h + v E a r t h 2 2 ( 13 c ) {\displaystyle (1-{\frac {2M}{r_{Earth}}}-v_{Earth}^{2})^{-1/2}\approx 1+{\frac {M}{r_{Earth}}}+{\frac {v_{Earth}^{2}}{2}}\quad \quad \quad \quad (13c)}
d t s a t l l i t e d t E a r t h ≈ ( 1 − M r s a t − v s a t 2 2 ) × ( 1 + M r E a r t h + v E a r t h 2 2 ) {\displaystyle {\frac {dt_{satllite}}{dt_{Earth}}}\approx \left(1-{\frac {M}{r_{sat}}}-{\frac {v_{sat}^{2}}{2}}\right)\times \left(1+{\frac {M}{r_{Earth}}}+{\frac {v_{Earth}^{2}}{2}}\right)}
= {\displaystyle =}
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![{\displaystyle t=-f\left[(n-1){\frac {df}{dt}}\right]^{-1}\left[1-\left({\frac {f}{f_{0}}}\right)^{n-1}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c04bcb6140e359217cf2dad657aaf5cc9719c2a8)
