1 Introduction
$$\begin{array}{c}{c}^{2}(t-{t}_{j}{)}^{2}=|\mathbf{r}-{\mathbf{r}}_{j}{|}^{2},j=1,2,3,4.\end{array}$$ | (1) |
2 Reference Frames and the Sagnac Effect
$$\begin{array}{c}-d{s}^{2}=-(cdt{)}^{2}+d{r}^{2}+{r}^{2}d{\phi}^{2}+d{z}^{2},\end{array}$$ | (2) |
$$\begin{array}{c}t={t}^{\prime},r={r}^{\prime},\phi ={\phi}^{\prime}+{\omega}_{E}{t}^{\prime},z={z}^{\prime}.\end{array}$$ | (3) |
$$\begin{array}{c}-d{s}^{2}=-\left(1-\frac{{\omega}_{E}^{2}{{r}^{\prime}}^{2}}{{c}^{2}}\right)(cd{t}^{\prime}{)}^{2}+2{\omega}_{E}{{r}^{\prime}}^{2}d{\phi}^{\prime}d{t}^{\prime}+(d{\sigma}^{\prime}{)}^{2},\end{array}$$ | (4) |
$$\begin{array}{c}(cd{t}^{\prime}{)}^{2}-\frac{2{\omega}_{E}{{r}^{\prime}}^{2}d{\phi}^{\prime}\left(cd{t}^{\prime}\right)}{c}-(d{\sigma}^{\prime}{)}^{2}=0,\end{array}$$ | (5) |
$$\begin{array}{c}cd{t}^{\prime}=d{\sigma}^{\prime}+\frac{{\omega}_{E}{{r}^{\prime}}^{2}d{\phi}^{\prime}}{c}.\end{array}$$ | (6) |
$$\begin{array}{c}{\int}_{path}d{t}^{\prime}={\int}_{path}\frac{d{\sigma}^{\prime}}{c}+\frac{2{\omega}_{E}}{{c}^{2}}{\int}_{path}d{A}_{z}^{\prime}.\left[\text{light}\right]\end{array}$$ | (7) |
$$\begin{array}{c}\frac{2{\omega}_{E}}{{c}^{2}}{\int}_{path}d{A}_{z}^{\prime}=207.4ns.\end{array}$$ | (8) |
$$\begin{array}{c}(d\tau {)}^{2}=(ds/c{)}^{2}=d{{t}^{\prime}}^{2}\left[1-{\left(\frac{{\omega}_{E}{r}^{\prime}}{c}\right)}^{2}-\frac{2{\omega}_{E}{{r}^{\prime}}^{2}d{\phi}^{\prime}}{{c}^{2}d{t}^{\prime}}-{\left(\frac{d{\sigma}^{\prime}}{cd{t}^{\prime}}\right)}^{2}\right].\end{array}$$ | (9) |
$$\begin{array}{c}d\tau =d{t}^{\prime}-\frac{{\omega}_{E}{{r}^{\prime}}^{2}d{\phi}^{\prime}}{{c}^{2}}\end{array}$$ | (10) |
$$\begin{array}{c}{\int}_{path}d{t}^{\prime}={\int}_{path}d\tau +\frac{2{\omega}_{E}}{{c}^{2}}{\int}_{path}d{A}_{z}^{\prime}.\left[\text{portable clock}\right]\end{array}$$ | (11) |
3 GPS Coordinate Time and TAI
$$\begin{array}{c}-d{s}^{2}=-\left(1+\frac{2V}{{c}^{2}}\right)(cdt{)}^{2}+\left(1-\frac{2V}{{c}^{2}}\right)(d{r}^{2}+{r}^{2}d{\theta}^{2}+{r}^{2}{sin}^{2}\theta d{\phi}^{2}).\end{array}$$ | (12) |
$$\begin{array}{c}V=-\frac{G{M}_{E}}{r}\left[1-{J}_{2}{\left(\frac{{a}_{1}}{r}\right)}^{2}{P}_{2}(cos\theta )\right].\end{array}$$ | (13) |
$$\begin{array}{c}t={t}^{\prime},r={r}^{\prime},\theta ={\theta}^{\prime},\phi ={\phi}^{\prime}+{\omega}_{E}{t}^{\prime}.\end{array}$$ | (14) |
$$\begin{array}{ccc}-d{s}^{2}=& -& \left[1+\frac{2V}{{c}^{2}}-{\left(\frac{{\omega}_{E}{r}^{\prime}sin{\theta}^{\prime}}{c}\right)}^{2}\right](cd{t}^{\prime}{)}^{2}+2{\omega}_{E}{{r}^{\prime}}^{2}{sin}^{2}{\theta}^{\prime}d{\phi}^{\prime}d{t}^{\prime}\end{array}$$ |
$$\begin{array}{ccc}& +& \left(1-\frac{2V}{{c}^{2}}\right)(d{{r}^{\prime}}^{2}+{{r}^{\prime}}^{2}d{{\theta}^{\prime}}^{2}+{{r}^{\prime}}^{2}{sin}^{2}{\theta}^{\prime}d{{\phi}^{\prime}}^{2}).\end{array}$$ | (15) |
$$\begin{array}{c}{g}_{00}^{\prime}=-\left[1+\frac{2V}{{c}^{2}}-{\left(\frac{{\omega}_{E}{r}^{\prime}sin{\theta}^{\prime}}{c}\right)}^{2}\right]\equiv -\left(1+\frac{2\Phi}{{c}^{2}}\right),\end{array}$$ | (16) |
$$\begin{array}{c}-d{s}^{2}=-\left(1+\frac{2V}{{c}^{2}}-\frac{{\omega}_{E}^{2}{{r}^{\prime}}^{2}{sin}^{2}{\theta}^{\prime}}{{c}^{2}}\right)(cd{t}^{\prime}{)}^{2},\end{array}$$ | (17) |
$$\begin{array}{ccc}\frac{{\Phi}_{0}}{{c}^{2}}& =& -\frac{G{M}_{E}}{{a}_{1}{c}^{2}}-\frac{G{M}_{E}{J}_{2}}{2{a}_{1}{c}^{2}}-\frac{{\omega}_{E}^{2}{a}_{1}^{2}}{2{c}^{2}}\end{array}$$ |
$$\begin{array}{ccc}& =& -6.95348\times {10}^{-10}-3.764\times {10}^{-13}-1.203\times {10}^{-12}\end{array}$$ |
$$\begin{array}{ccc}& =& -6.96927\times {10}^{-10}.\end{array}$$ | (18) |
$$\begin{array}{c}d\tau =ds/c=d{t}^{\prime}\left(1+\frac{{\Phi}_{0}}{{c}^{2}}\right).\end{array}$$ | (19) |
$$\begin{array}{c}{r}^{\prime}=(6356742.025+21353.642{{x}^{\prime}}^{2}+39.832{{x}^{\prime}}^{4}+0.798{{x}^{\prime}}^{6}+0.003{{x}^{\prime}}^{8})m.\end{array}$$ | (20) |
$$\begin{array}{c}{t}^{\prime \prime}=(1+{\Phi}_{0}/{c}^{2}){t}^{\prime}=(1+{\Phi}_{0}/{c}^{2})t.\end{array}$$ | (21) |
$$\begin{array}{ccc}-d{s}^{2}=& -& \left(1+\frac{2(\Phi -{\Phi}_{0})}{{c}^{2}}\right)(cd{t}^{\prime \prime}{)}^{2}+2{\omega}_{E}{{r}^{\prime}}^{2}{sin}^{2}{\theta}^{\prime}d{\phi}^{\prime}d{t}^{\prime \prime}\end{array}$$ |
$$\begin{array}{ccc}& +& \left(1-\frac{2V}{{c}^{2}}\right)(d{{r}^{\prime}}^{2}+{{r}^{\prime}}^{2}d{{\theta}^{\prime}}^{2}+{{r}^{\prime}}^{2}{sin}^{2}{\theta}^{\prime}d{{\phi}^{\prime}}^{2}),\end{array}$$ | (22) |
$$\begin{array}{c}-d{s}^{2}=-\left(1+\frac{2(V-{\Phi}_{0})}{{c}^{2}}\right)(cd{t}^{\prime \prime}{)}^{2}+\left(1-\frac{2V}{{c}^{2}}\right)(d{r}^{2}+{r}^{2}d{\theta}^{2}+{r}^{2}{sin}^{2}\theta d{\phi}^{2}).\end{array}$$ | (23) |
^{ $\text{1}$ } WGS-84(G873) values of these constants are used in this article.
4 The Realization of Coordinate Time
$$\begin{array}{c}-d{s}^{2}=-\left(1+\frac{2(V-{\Phi}_{0})}{{c}^{2}}\right)(cdt{)}^{2}+\left(1-\frac{2V}{{c}^{2}}\right)(d{r}^{2}+{r}^{2}d{\theta}^{2}+{r}^{2}{sin}^{2}\theta d{\phi}^{2}).\end{array}$$ | (24) |
$$\begin{array}{c}-d{s}^{2}=-\left[1+\frac{2(V-{\Phi}_{0})}{{c}^{2}}-\left(1-\frac{2V}{{c}^{2}}\right)\frac{d{r}^{2}+{r}^{2}d{\theta}^{2}+{r}^{2}{sin}^{2}\theta d{\phi}^{2}}{(cdt{)}^{2}}\right](cdt{)}^{2}.\end{array}$$ | (25) |
$$\begin{array}{c}{v}^{2}=\frac{d{r}^{2}+{r}^{2}d{\theta}^{2}+{r}^{2}{sin}^{2}\theta d{\phi}^{2}}{d{t}^{2}}.\end{array}$$ | (26) |
$$\begin{array}{c}d\tau =ds/c=\left[1+\frac{(V-{\Phi}_{0})}{{c}^{2}}-\frac{{v}^{2}}{2{c}^{2}}\right]dt.\end{array}$$ | (27) |
$$\begin{array}{c}{\int}_{path}dt={\int}_{path}d\tau \left[1-\frac{(V-{\Phi}_{0})}{{c}^{2}}-\frac{{v}^{2}}{2{c}^{2}}\right].\end{array}$$ | (28) |
5 Relativistic Effects on Satellite Clocks
$$\begin{array}{c}r=a(1-{e}^{2})/(1+ecosf).\end{array}$$ | (29) |
$$\begin{array}{c}\begin{array}{ccc}cosf& =& \frac{cosE-e}{1-ecosE},\\ \\ sinf& =& \sqrt{1-{e}^{2}}\frac{sinE}{1-ecosE}.\end{array}\end{array}$$ | (30) |
$$\begin{array}{c}r=a(1-ecosE).\end{array}$$ | (31) |
$$\begin{array}{c}E-esinE=\sqrt{\frac{G{M}_{E}}{{a}^{3}}}(t-{t}_{p}),\end{array}$$ | (32) |
$$\begin{array}{c}\frac{1}{2}{v}^{2}-\frac{G{M}_{E}}{r}=-\frac{G{M}_{E}}{2a}.\end{array}$$ | (33) |
$$\begin{array}{c}\Delta t={\int}_{path}d\tau \left[1+\frac{3G{M}_{E}}{2a{c}^{2}}+\frac{{\Phi}_{0}}{{c}^{2}}-\frac{2G{M}_{E}}{{c}^{2}}\left(\frac{1}{a}-\frac{1}{r}\right)\right].\end{array}$$ | (34) |
$$\begin{array}{c}\frac{3G{M}_{E}}{2a{c}^{2}}+\frac{{\Phi}_{0}}{{c}^{2}}=+2.5046\times {10}^{-10}-6.9693\times {10}^{-10}=-4.4647\times {10}^{-10}.\end{array}$$ | (35) |
$$\begin{array}{c}\left[1-4.4647\times {10}^{-10}\right]\times 10.23MHz=10.22999999543MHz.\end{array}$$ | (36) |
$$\begin{array}{c}\frac{dE}{dt}=\frac{\sqrt{G{M}_{E}/{a}^{3}}}{1-ecosE}.\end{array}$$ | (37) |
$$\begin{array}{ccc}\int \left[\frac{2G{M}_{E}}{{c}^{2}}\left(\frac{1}{r}-\frac{1}{a}\right)\right]\frac{ds}{c}& \simeq & \frac{2G{M}_{E}}{{c}^{2}}\int \left(\frac{1}{r}-\frac{1}{a}\right)dt\end{array}$$ |
$$\begin{array}{ccc}& =& \frac{2G{M}_{E}}{a{c}^{2}}\int dt\left(\frac{ecosE}{1-ecosE}\right)\end{array}$$ |
$$\begin{array}{ccc}& =& \frac{2\sqrt{G{M}_{E}a}}{{c}^{2}}e\left(sinE-sin{E}_{0}\right)\end{array}$$ |
$$\begin{array}{ccc}& =& +\frac{2\sqrt{G{M}_{E}a}}{{c}^{2}}esinE+constant.\end{array}$$ | (38) |
$$\begin{array}{c}\Delta {t}_{r}=+4.4428\times {10}^{-10}e\sqrt{a}sinE\frac{s}{\sqrt{m}}.\end{array}$$ | (39) |
$$\begin{array}{c}\Delta {t}_{r}=+\frac{2\mathbf{r}\cdot \mathbf{v}}{{c}^{2}},\end{array}$$ | (40) |
6 TOPEX/POSEIDON Relativity Experiment
$$\begin{array}{c}{\rho}_{j}=c({t}_{R}^{\prime}-{t}_{j}^{\prime})=c\left[({t}_{R}+{b}_{R})-({t}_{j}+{b}_{j}-(\Delta {t}_{r}{)}_{j})\right].\end{array}$$ | (41) |
$$\begin{array}{c}\left|{\mathbf{r}}_{R}\right({t}_{R})-{\mathbf{r}}_{j}({t}_{j}\left)\right|=c({t}_{R}-{t}_{j}).\end{array}$$ | (42) |
$$\begin{array}{c}\left|{\mathbf{r}}_{R}\right({t}_{R})-{\mathbf{r}}_{j}({t}_{j}\left)\right|-{\rho}_{j}+c{b}_{R}-c{b}_{j}+c(\Delta {t}_{r}{)}_{j}=0.\end{array}$$ | (43) |
$$\begin{array}{c}\left|{\mathbf{r}}_{R}\right({t}_{R})-{\mathbf{r}}_{j}({t}_{j}\left)\right|-{\rho}_{j}-c{b}_{j}+c\Delta {t}_{r}=-c{b}_{R}.\end{array}$$ | (44) |
$$\begin{array}{c}\left|{\mathbf{r}}_{R}\right({t}_{R})-{\mathbf{r}}_{j}({t}_{j}\left)\right|-{\rho}_{j}-c{b}_{j}+c{b}_{R}=-c\Delta {t}_{r}.\end{array}$$ | (45) |
7 Doppler Effect
$$\begin{array}{c}{f}_{R}={f}_{0}\left[1+\frac{-{V}_{R}+{\mathbf{v}}_{R}^{2}/2+{\Phi}_{0}+2G{M}_{E}/a+2{V}_{j}}{{c}^{2}}\right]\frac{\left(1-\mathbf{N}\cdot {\mathbf{v}}_{R}/c\right)}{\left(1-\mathbf{N}\cdot {\mathbf{v}}_{j}/c\right)},\end{array}$$ | (46) |
$$\begin{array}{c}{f}_{R}={f}_{0}\left[1+\frac{2G{M}_{E}}{{c}^{2}}\left(\frac{1}{a}-\frac{1}{r}\right)\right]\frac{\left(1-\mathbf{N}\cdot {\mathbf{v}}_{R}/c\right)}{\left(1-\mathbf{N}\cdot {\mathbf{v}}_{j}/c\right)}.\end{array}$$ | (47) |
8 Crosslink Ranging
$$\begin{array}{c}{T}^{\left(i\right)}={T}_{S}^{\left(i\right)}+\frac{2\sqrt{GM{a}_{i}}}{{c}^{2}}{e}_{i}sin{E}_{i}.\end{array}$$ | (48) |
$$\begin{array}{c}{T}_{S}^{\left(j\right)}={T}^{\left(j\right)}-\frac{2\sqrt{GM{a}_{j}}}{{c}^{2}}{e}_{j}sin{E}_{j}.\end{array}$$ | (49) |
$$\begin{array}{c}{T}_{S}^{\left(j\right)}={T}_{S}^{\left(i\right)}+\frac{l}{c}-\frac{2\sqrt{GM{a}_{j}}}{{c}^{2}}{e}_{j}sin{E}_{j}+\frac{2\sqrt{GM{a}_{i}}}{{c}^{2}}{e}_{i}sin{E}_{i}.\end{array}$$ | (50) |
$$\begin{array}{c}l=|\Delta \mathbf{r}|+\frac{\Delta \mathbf{r}\cdot {\mathbf{v}}_{j}}{c}.\end{array}$$ | (51) |
$$\begin{array}{c}{T}_{S}^{\left(j\right)}={T}_{S}^{\left(i\right)}+\frac{|\Delta \mathbf{r}|}{c}-\frac{2\sqrt{GM{a}_{2}}}{{c}^{2}}{e}_{j}sin{E}_{j}+\frac{2\sqrt{GM{a}_{i}}}{{c}^{2}}{e}_{i}sin{E}_{i}+\frac{\Delta \mathbf{r}\cdot {\mathbf{v}}_{j}}{{c}^{2}}.\end{array}$$ | (52) |
9 Frequency Shifts Induced by Orbit Changes
$$\begin{array}{c}\frac{\Delta f}{f}=-\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}-\frac{G{M}_{E}}{r{c}^{2}}-\frac{{\Phi}_{0}}{{c}^{2}},\end{array}$$ | (53) |
$$\begin{array}{c}\frac{\Delta f}{f}=-\frac{3G{M}_{E}}{2a{c}^{2}}-\frac{{\Phi}_{0}}{{c}^{2}}+\frac{2G{M}_{E}}{{c}^{2}}\left[\frac{1}{r}-\frac{1}{a}\right].\end{array}$$ | (54) |
$$\begin{array}{c}\frac{\delta f}{f}=+\frac{3G{M}_{E}\delta a}{2{c}^{2}{a}^{2}}.\end{array}$$ | (55) |
$$\begin{array}{c}\frac{\delta f}{f}=-1.85\times {10}^{-13}\left(\text{measured}\right).\end{array}$$ | (56) |
$$\begin{array}{ccc}07/22/00:a& =& 2.656139556\times {10}^{7}m;v=3.873947951\times {10}^{3}m{s}^{-1},\end{array}$$ |
$$\begin{array}{ccc}07/30/00:a& =& 2.654267359\times {10}^{7}m;v=3.875239113\times {10}^{3}m{s}^{-1}.\end{array}$$ |
$$\begin{array}{c}\frac{\delta f}{f}=-1.734\times {10}^{-13},\end{array}$$ | (57) |
$$\begin{array}{c}V(x,y,z)=-\frac{G{M}_{E}}{r}\left(1-\frac{{J}_{2}{a}_{1}^{2}}{{r}^{2}}\left[\frac{3{z}^{2}}{2{r}^{2}}-\frac{1}{2}\right]\right).\end{array}$$ | (58) |
$$\begin{array}{c}\epsilon =constant=\frac{{v}^{2}}{2}+V(x,y,z)=\frac{{v}^{2}}{2}-\frac{G{M}_{E}}{r}+{V}^{\prime}(x,y,z),\end{array}$$ | (59) |
$$\begin{array}{c}\epsilon =-\frac{G{M}_{E}}{2a}+{V}^{\prime}(x,y,z),\end{array}$$ | (60) |
$$\begin{array}{c}\frac{{v}^{2}}{2}-\frac{G{M}_{E}}{r}=-\frac{G{M}_{E}}{2a}.\end{array}$$ | (61) |
$$\begin{array}{c}\frac{\Delta f}{f}=-\frac{{v}^{2}}{2}-\frac{G{M}_{E}}{r}+{V}^{\prime}(x,y,z).\end{array}$$ | (62) |
$$\begin{array}{c}E-esinE=M={n}_{0}(t-{t}_{0}),\end{array}$$ | (63) |
$$\begin{array}{c}{n}_{0}=\sqrt{G{M}_{E}/{a}^{3}}.\end{array}$$ | (64) |
$$\begin{array}{c}r=a(1-ecosE),\end{array}$$ | (65) |
$$\begin{array}{c}cosf=\frac{cosE-e}{1-ecosE},sinf=\sqrt{1-{e}^{2}}\frac{sinE}{1-ecosE}.\end{array}$$ | (66) |
$$\begin{array}{ccc}x& =& r(cos\Omega cos(f+\omega )-cosisin\Omega sin(f+\omega \left)\right),\end{array}$$ | (67) |
$$\begin{array}{ccc}y& =& r(sin\Omega cos(f+\omega )+cosicos\Omega sin(f+\omega \left)\right),\end{array}$$ | (68) |
$$\begin{array}{ccc}z& =& r(sinisin(f+\omega \left)\right),\end{array}$$ | (69) |
$$\begin{array}{c}{v}^{2}=\frac{G{M}_{E}}{a}\frac{1+ecosE}{1-ecosE}.\end{array}$$ | (70) |
$$\begin{array}{ccc}e={e}_{0}+\frac{3{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{2}}[& & \left(1-\frac{3}{2}sin2i0\right)cosf+\frac{1}{4}{sin}^{2}{i}_{0}cos(2{\omega}_{0}+f)\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{7}{12}{sin}^{2}{i}_{0}cos(2{\omega}_{0}+3f)],\end{array}$$ | (71) |
$$\begin{array}{c}E=M+esinE,\end{array}$$ | (72) |
$$\begin{array}{c}cosE=cosM-esinMsinE,\end{array}$$ | (73) |
$$\begin{array}{c}ecosE=ecosM-{e}^{2}sinMsinE\approx ecosM.\end{array}$$ | (74) |
$$\begin{array}{c}M={M}_{0}+\Delta M/{e}_{0},\end{array}$$ | (75) |
$$\begin{array}{ccc}\Delta M/{e}_{0}=-\frac{3{J}_{2}{a}_{1}^{2}}{2{e}_{0}{a}_{0}^{2}}[& & \left(1-\frac{3}{2}sin2i0\right)sinf-\frac{1}{4}{sin}^{2}{i}_{0}sin(2{\omega}_{0}+f)\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{7}{12}{sin}^{2}{i}_{0}sin(2{\omega}_{0}+3f)],\end{array}$$ | (76) |
$$\begin{array}{c}ecosE=ecos{M}_{0}-\Delta Msin{M}_{0}.\end{array}$$ | (77) |
$$\begin{array}{c}ecosE={e}_{0}cos{E}_{0}+\frac{3{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{2}}\left(1-\frac{3}{2}sin2i0\right)+\frac{5{J}_{2}{a}_{1}^{2}}{4{a}_{0}^{2}}{sin}^{2}{i}_{0}cos2({\omega}_{0}+f),\end{array}$$ | (78) |
$$\begin{array}{c}a={a}_{0}+\frac{3{J}_{2}{a}_{1}^{2}}{2{a}_{0}}{sin}^{2}{i}_{0}cos2({\omega}_{0}+f),\end{array}$$ | (79) |
$$\begin{array}{ccc}r& =& {a}_{0}(1-{e}_{0}cos{E}_{0})+\Delta a-\Delta (ecosE)\end{array}$$ |
$$\begin{array}{ccc}& =& {a}_{0}(1-{e}_{0}cos{E}_{0})-\frac{3{J}_{2}{a}_{1}^{2}}{2{a}_{0}}\left(1-\frac{3}{2}sin2i0\right)\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{{J}_{2}{a}_{1}^{2}}{4{a}_{0}}{sin}^{2}{i}_{0}cos2({\omega}_{0}+f).\end{array}$$ | (80) |
$$\begin{array}{ccc}\frac{{v}^{2}}{2}& =& \frac{G{M}_{E}}{2{a}_{0}}\left(1+2{e}_{0}cos{E}_{0}\right)+\frac{3G{M}_{E}{J}_{2}{a}_{1}^{2}}{{a}_{0}^{3}}\left(1-\frac{3}{2}sin2i0\right)\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{3}}{sin}^{2}{i}_{0}cos2({\omega}_{0}+f).\end{array}$$ | (81) |
$$\begin{array}{ccc}-\frac{G{M}_{E}}{r}& =& -\frac{G{M}_{E}}{{a}_{0}}(1+{e}_{0}cos{E}_{0})-\frac{3G{M}_{E}{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{3}}\left(1-\frac{3}{2}sin2i0\right)\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{4{a}_{0}^{3}}cos2({\omega}_{0}+f).\end{array}$$ | (82) |
$$\begin{array}{c}{V}^{\prime}(x,y,z)=-\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{3}}\left(1-\frac{3}{2}sin2i0\right)-\frac{3G{M}_{E}{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{4{a}_{0}^{3}}cos2({\omega}_{0}+f).\end{array}$$ | (83) |
$$\begin{array}{c}\epsilon =\frac{{v}^{2}}{2}-\frac{G{M}_{E}}{r}+{V}^{\prime}=-\frac{G{M}_{E}}{2{a}_{0}}-\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{3}}\left(1-\frac{3}{2}sin2i0\right).\end{array}$$ | (84) |
$$\begin{array}{ccc}\frac{\Delta f}{f}& =& -\frac{{v}^{2}}{2{c}^{2}}-\frac{G{M}_{E}}{{c}^{2}r}+\frac{{V}^{\prime}}{{c}^{2}}\end{array}$$ |
$$\begin{array}{ccc}& =& -\frac{3G{M}_{E}}{2{a}_{0}{c}^{2}}-\frac{2G{M}_{E}}{{c}^{2}{a}_{0}}{e}_{0}cos{E}_{0}-\frac{7G{M}_{E}{J}_{2}{a}_{1}^{2}}{2{a}_{0}^{3}{c}^{2}}\left(1-\frac{3}{2}sin2i0\right)\end{array}$$ |
$$\begin{array}{ccc}& & -\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{{a}_{0}^{3}{c}^{2}}cos2({\omega}_{0}+f).\end{array}$$ | (85) |
$$\begin{array}{c}\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{{a}_{0}^{3}{c}^{2}}=6.95\times {10}^{-15},\end{array}$$ | (86) |
$$\begin{array}{c}\frac{\Delta f}{f}=3\epsilon /{c}^{2}.\end{array}$$ | (87) |
$$\begin{array}{c}\delta {t}_{{J}_{2}}={\int}_{path}dt\left[-\frac{G{M}_{E}{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{{a}_{0}^{3}{c}^{2}}cos(2{\omega}_{0}+2nt)\right],\end{array}$$ | (88) |
$$\begin{array}{c}\delta {t}_{{J}_{2}}=-\sqrt{\frac{G{M}_{E}}{{a}_{0}^{3}}}\frac{{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{2{c}^{2}}sin(2{\omega}_{0}+2nt).\end{array}$$ | (89) |
$$\begin{array}{c}\delta {t}_{{J}_{2}}(correction)=\sqrt{\frac{G{M}_{E}}{{a}_{0}^{3}}}\frac{{J}_{2}{a}_{1}^{2}{sin}^{2}{i}_{0}}{2{c}^{2}}sin(2{\omega}_{0}+2nt).\end{array}$$ | (90) |
$$\begin{array}{ccc}07/22/00:a& =& (2.65611575\times {10}^{7}\pm 69)m,\end{array}$$ |
$$\begin{array}{ccc}07/30/00:a& =& (2.65423597\times {10}^{3}\pm 188)m,\end{array}$$ |
$$\begin{array}{ccc}10/07/00:a& =& (2.65418742\times {10}^{7}\pm 95)m,\end{array}$$ |
$$\begin{array}{ccc}10/12/00:a& =& (2.65606323\times {10}^{7}\pm 58)m.\end{array}$$ |
$$\begin{array}{c}\frac{\Delta f}{f}=-1.77\times {10}^{-13},\end{array}$$ | (91) |
$$\begin{array}{c}\frac{\Delta f}{f}=+1.75\times {10}^{-13}.\end{array}$$ | (92) |
$$\begin{array}{ccc}03/07/01:a& =& (2.65597188\times {10}^{7}\pm 140)m,\end{array}$$ |
$$\begin{array}{ccc}03/11/01:a& =& (2.65359261\times {10}^{7}\pm 108)m.\end{array}$$ |
$$\begin{array}{c}\frac{\Delta f}{f}=-2.24\times {10}^{-13}\pm 0.02\times {10}^{-13}.\end{array}$$ | (93) |
$$\begin{array}{c}\frac{\delta f}{f}=+\frac{3G{M}_{E}\delta a}{2{c}^{2}{a}^{2}}.\end{array}$$ | (94) |
10 Secondary Relativistic Effects
$$\begin{array}{c}dt=\frac{1}{c}\left[1-\frac{2V}{{c}^{2}}+\frac{{\Phi}_{0}}{{c}^{2}}\right]d\sigma .\end{array}$$ | (95) |
$$\begin{array}{c}\Delta {t}_{delay}=\frac{{\Phi}_{0}}{{c}^{2}}\frac{l}{c}+\frac{2G{M}_{E}}{{c}^{3}}ln\left[\frac{{r}_{1}+{r}_{2}+l}{{r}_{1}+{r}_{2}-l}\right],\end{array}$$ | (96) |
$$\begin{array}{c}{\int}_{{r}_{1}}^{{r}_{2}}dr\left[1+\frac{G{M}_{E}}{{c}^{2}r}\right]={r}_{2}-{r}_{1}+\frac{G{M}_{E}}{{c}^{2}}ln\left(\frac{{r}_{2}}{{r}_{1}}\right).\end{array}$$ | (97) |
11 Applications
12 Conclusions
Note: The reference version of this article is published by Living Reviews in Relativity