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<title>Gauss's method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Gauss's method</span></span>

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<p>In <a href="Orbital_mechanics" title="Orbital mechanics">orbital mechanics</a> (a subfield of <a href="Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>), <b>Gauss's method</b> is used for preliminary <a href="Orbit_determination" title="Orbit determination">orbit determination</a> from at least three observations (more observations increases the accuracy of the determined orbit) of the orbiting body of interest at three different times. The required information are the times of observations, the position vectors of the observation points (in <a href="Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial Coordinate System</a>), the <a href="Direction_cosine" title="Direction cosine">direction cosine</a> vector of the orbiting body from the observation points (from Topocentric Equatorial Coordinate System) and general physical data.

</p><p>Working in 1801, <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> developed important mathematical techniques (summed up in Gauss's methods) which were specifically used to determine the orbit of <a href="Ceres_(dwarf_planet)" title="Ceres (dwarf planet)">Ceres</a>. The method shown following is the orbit determination of an orbiting body about the focal body where the observations were taken from, whereas the method for determining Ceres' orbit requires a bit more effort because the observations were taken from <a href="Earth" title="Earth">Earth</a> while Ceres orbits the <a href="Sun" title="Sun">Sun</a>.

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<div class="mw-heading mw-heading2"><h2 id="Observer_position_vector">Observer position vector</h2></div>

<p>The observer position vector (in <a href="Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial coordinate system</a>) of the observation points can be determined from the <a href="Latitude" title="Latitude">latitude</a> and <a href="Local_sidereal_time" class="mw-redirect" title="Local sidereal time">local sidereal time</a> (from <a href="Topocentric_coordinate_system" class="mw-redirect" title="Topocentric coordinate system">Topocentric coordinate system</a>) at the surface of the focal body of the orbiting body (for example, the Earth) via either:

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:164px;max-width:164px"><div class="trow"><div class="tsingle" style="width:162px;max-width:162px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Geodetic latitude</div></div></div><div class="trow"><div class="tsingle" style="width:162px;max-width:162px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Geocentric latitude</div></div></div></div></div>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {R_{n}} &=\left[{R_{e} \over {\sqrt {1-(2f-f^{2})\sin ^{2}\phi _{n}}}}+H_{n}\right]\cos \phi _{n}(\cos \theta _{n}\ \mathbf {\hat {I}} +\sin \theta _{n}\ \mathbf {\hat {J}} )+\left[{R_{e}(1-f)^{2} \over {\sqrt {1-(2f-f^{2})\sin ^{2}\phi _{n}}}}+H_{n}\right]\sin \phi _{n}\ \mathbf {\hat {K}} \\&=\left[{R_{e} \over {\sqrt {1-e^{2}\sin ^{2}\phi _{n}}}}+H_{n}\right]\cos \phi _{n}(\cos \theta _{n}\ \mathbf {\hat {I}} +\sin \theta _{n}\ \mathbf {\hat {J}} )+\left[{R_{e}(1-e^{2}) \over {\sqrt {1-e^{2}\sin ^{2}\phi _{n}}}}+H_{n}\right]\sin \phi _{n}\ \mathbf {\hat {K}} \end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{n}} =r_{e}\cos \phi '_{n}\cos \theta _{n}\ \mathbf {\hat {I}} +r_{e}\cos \phi '_{n}\sin \theta _{n}\ \mathbf {\hat {J}} +r_{e}\sin \phi '_{n}\ \mathbf {\hat {K}} }</annotation>

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where,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{n}} }</annotation>

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</math></span><img src="./2a7bf364ea5d9bd5b4258eccedae9057b76bfcde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.286ex; height:2.509ex;" alt="{\displaystyle \mathbf {R_{n}} }" loading="lazy"></span> is the respective observer position vector (in Equatorial Coordinate System)</li>

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<annotation encoding="application/x-tex">{\displaystyle R_{e}}</annotation>

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</math></span><img src="./c0db90f44b6b50f5e049da49e2dea514a260ba9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.509ex;" alt="{\displaystyle R_{e}}" loading="lazy"></span> is the equatorial radius of the central body (e.g., 6,378 km for Earth)</li>

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<annotation encoding="application/x-tex">{\displaystyle r_{e}}</annotation>

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</math></span><img src="./c652979f43cb2855b9a028560d7e228aa2f9bb1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.047ex; height:2.009ex;" alt="{\displaystyle r_{e}}" loading="lazy"></span> is the geocentric distance</li>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is the oblateness (or <a href="Flattening" title="Flattening">flattening</a>) of the central body (e.g., 0.003353 for Earth)</li>

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> is the eccentricity of the central body (e.g., 0.081819 for Earth)</li>

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<annotation encoding="application/x-tex">{\displaystyle \phi '_{n}}</annotation>

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</math></span><img src="./6dfdf0f5e119f14b20bd92793003b6e205621aa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.604ex; height:2.509ex;" alt="{\displaystyle \phi '_{n}}" loading="lazy"></span> is the <a href="Latitude#Geocentric_latitude" title="Latitude">geocentric latitude</a> (the angle between the radius and the equatorial plane)</li>

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<annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation>

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</math></span><img src="./e63458b04288bbe116a9a8037dfae0b36b2c639a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}" loading="lazy"></span> is the <a href="Geodetic_altitude" class="mw-redirect" title="Geodetic altitude">geodetic altitude</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \theta _{n}}</annotation>

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</math></span><img src="./79cc00920259451fc1a684ba7350b6f93ce4f08a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.509ex;" alt="{\displaystyle \theta _{n}}" loading="lazy"></span> is the <a href="Local_sidereal_time" class="mw-redirect" title="Local sidereal time">local sidereal time</a> of observation site</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Orbiting_body_direction_cosine_vector">Orbiting body direction cosine vector</h2></div>

<p>The orbiting body direction cosine vector can be determined from the <a href="Right_ascension" title="Right ascension">right ascension</a> and <a href="Declination" title="Declination">declination</a> (from Topocentric Equatorial Coordinate System) of the orbiting body from the observation points via:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} =\cos \delta _{n}\cos \alpha _{n}\ \mathbf {\hat {I}} +\cos \delta _{n}\sin \alpha _{n}\ \mathbf {\hat {J}} +\sin \delta _{n}\ \mathbf {\hat {K}} }</annotation>

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where,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }</annotation>

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</math></span><img src="./bd7bce4ed0ef2dcef5551ecd72b7e8f11a9f3b31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.885ex; height:2.843ex;" alt="{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }" loading="lazy"></span> is the respective <a href="Unit_vector" title="Unit vector">unit vector</a> in the direction of the position vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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</math></span><img src="./50acda784849200b94d62c19614b5e143adf169e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle \delta _{n}}" loading="lazy"></span> is the respective declination</li>

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<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

<p>The initial derivation begins with vector addition to determine the orbiting body's position vector. Then based on the conservation of <a href="Angular_momentum" title="Angular momentum">angular momentum</a> and <a href="Keplerian_orbit" class="mw-redirect" title="Keplerian orbit">Keplerian orbit</a> principles (which states that an orbit lies in a two dimensional plane in three dimensional space), a linear combination of said position vectors is established. Also, the relation between a body's position and velocity vector by <a href="Lagrange_polynomials" class="mw-redirect" title="Lagrange polynomials">Lagrange coefficients</a> is used which results in the use of said coefficients. Then with vector manipulation and algebra, the following equations were derived. For detailed derivation, refer to Curtis.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>NOTE: Gauss's method is a preliminary orbit determination, with emphasis on preliminary. The approximation of the Lagrange coefficients and the limitations of the required observation conditions (i.e., insignificant curvature in the arc between observations, refer to Gronchi<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> for more details) causes inaccuracies. Gauss's method can be improved, however, by increasing the accuracy of sub-components, such as solving <a href="Kepler's_equation" title="Kepler's equation">Kepler's equation</a>. Another way to increase the accuracy is through more observations.

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<p>Calculate time intervals, subtract the times between observations:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tau _{1}&=t_{1}-t_{2}\\\tau _{3}&=t_{3}-t_{2}\\\tau &=t_{3}-t_{1}\end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation>

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</math></span><img src="./271566db7e8ca8616a4dc3efb6c5982a2d987ee3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}" loading="lazy"></span> is the respective observation time</li></ul>

<p>Calculate cross products, take the cross products of the observational unit direction (order matters):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {p_{1}} &=\mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \times \mathbf {{\hat {\boldsymbol {\rho }}}_{3}} \\\mathbf {p_{2}} &=\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \times \mathbf {{\hat {\boldsymbol {\rho }}}_{3}} \\\mathbf {p_{3}} &=\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \times \mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \end{aligned}}}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} {\text{ and }}\mathbf {b} }</annotation>

</semantics>

</math></span><img src="./b4ae9d9d1f8531d76c9f70616a7d031233e368ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.694ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} {\text{ and }}\mathbf {b} }" loading="lazy"></span></li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p_{n}} }</annotation>

</semantics>

</math></span><img src="./084d36473d5c430a26dd8e47e182080e2cc917a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.768ex; height:2.009ex;" alt="{\displaystyle \mathbf {p_{n}} }" loading="lazy"></span> is the respective cross product vector</li>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }</annotation>

</semantics>

<p>Calculate common scalar quantity (scalar triple product), take the dot product of the first observational unit vector with the cross product of the second and third observational unit vector:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}=\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \cdot \mathbf {p_{1}} =\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \cdot (\mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \times \mathbf {{\hat {\boldsymbol {\rho }}}_{3}} )}">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}=\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \cdot \mathbf {p_{1}} =\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \cdot (\mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \times \mathbf {{\hat {\boldsymbol {\rho }}}_{3}} )}</annotation>

</semantics>

</math></span></span>

where

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} }</annotation>

</semantics>

</math></span><img src="./494aed3b5e94f1c0ee071debc707d2700c0e0390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.464ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} }" loading="lazy"></span> is the <a href="Dot_product" title="Dot product">dot product</a> of vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation>

</semantics>

</math></span><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation>

</semantics>

</math></span><img src="./13ebf4628a1adf07133a6009e4a78bdd990c6eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }" loading="lazy"></span></li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation>

</semantics>

</math></span><img src="./b943acf3ade50743482c4ee4570ad3de335d90d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{0}}" loading="lazy"></span> is the common scalar <a href="Triple_product" title="Triple product">triple product</a></li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p_{n}} }</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }</annotation>

</semantics>

<p>Calculate nine scalar quantities (similar to step 3):

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D_{11}&=\mathbf {R_{1}} \cdot \mathbf {p_{1}} &D_{12}&=\mathbf {R_{1}} \cdot \mathbf {p_{2}} &D_{13}&=\mathbf {R_{1}} \cdot \mathbf {p_{3}} \\D_{21}&=\mathbf {R_{2}} \cdot \mathbf {p_{1}} &D_{22}&=\mathbf {R_{2}} \cdot \mathbf {p_{2}} &D_{23}&=\mathbf {R_{2}} \cdot \mathbf {p_{3}} \\D_{31}&=\mathbf {R_{3}} \cdot \mathbf {p_{1}} &D_{32}&=\mathbf {R_{3}} \cdot \mathbf {p_{2}} &D_{33}&=\mathbf {R_{3}} \cdot \mathbf {p_{3}} \end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{mn}}</annotation>

</semantics>

</math></span><img src="./dec02033baee545bfc88bea8536c0eec47d5f2be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.586ex; height:2.509ex;" alt="{\displaystyle D_{mn}}" loading="lazy"></span> is the respective scalar quantity</li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{m}} }</annotation>

</semantics>

</math></span><img src="./003bb1155134ad2a212293854b87d11adaf84004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.81ex; height:2.509ex;" alt="{\displaystyle \mathbf {R_{m}} }" loading="lazy"></span> is the respective observer position vector</li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p_{n}} }</annotation>

</semantics>

<p>Calculate scalar position coefficients:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A&={\frac {1}{D_{0}}}\left(-D_{12}{\frac {\tau _{3}}{\tau }}+D_{22}+D_{32}{\frac {\tau _{1}}{\tau }}\right)\\B&={\frac {1}{6D_{0}}}\left[D_{12}\left(\tau _{3}^{2}-\tau ^{2}\right){\frac {\tau _{3}}{\tau }}+D_{32}\left(\tau ^{2}-\tau _{1}^{2}\right){\frac {\tau _{1}}{\tau }}\right]\\E&=\mathbf {R_{2}} \cdot \mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \end{aligned}}}">

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&={\frac {1}{D_{0}}}\left(-D_{12}{\frac {\tau _{3}}{\tau }}+D_{22}+D_{32}{\frac {\tau _{1}}{\tau }}\right)\\B&={\frac {1}{6D_{0}}}\left[D_{12}\left(\tau _{3}^{2}-\tau ^{2}\right){\frac {\tau _{3}}{\tau }}+D_{32}\left(\tau ^{2}-\tau _{1}^{2}\right){\frac {\tau _{1}}{\tau }}\right]\\E&=\mathbf {R_{2}} \cdot \mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> are scalar position coefficients</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{mn}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{n}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{n}}</annotation>

</semantics>

</math></span><img src="./08a51eb87e8de827a6df940f756f9ab254cb336b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle R_{n}}" loading="lazy"></span> is the respective observer position vector</li>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }</annotation>

</semantics>

<p>Calculate the squared scalar distance of the second observation, by taking the dot product of the position vector of the second observation:

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {R_{2}}^{2}=\mathbf {R_{2}} \cdot \mathbf {R_{2}} }</annotation>

</semantics>

</math></span></span>

where

</p>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {R_{2}}^{2}}</annotation>

</semantics>

</math></span><img src="./60f5318ace6cbefbb091d27834d8d0d811e7b8ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.872ex; height:3.009ex;" alt="{\displaystyle {R_{2}}^{2}}" loading="lazy"></span> is the squared distance of the second observation</li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{2}} }</annotation>

</semantics>

</math></span><img src="./66387413f43b61d09b028428c5fad532b90ae51c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.181ex; height:2.509ex;" alt="{\displaystyle \mathbf {R_{2}} }" loading="lazy"></span> is the position vector of the second observation</li></ul>

<p>Calculate the coefficients of the scalar distance polynomial for the second observation of the orbiting body:

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=-\left(A^{2}+2AE+{R_{2}}^{2}\right)\\b&=-2\mu B(A+E)\\c&=-\mu ^{2}B^{2}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a{\text{, }}b{\text{, and }}c}</annotation>

</semantics>

</math></span><img src="./86321d2bde421ec85e8f343f6f54ad9bf6e0f0fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.017ex; height:2.509ex;" alt="{\displaystyle a{\text{, }}b{\text{, and }}c}" loading="lazy"></span> are coefficients of the scalar distance polynomial for the second observation of the orbiting body</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A{\text{, }}B{\text{, and }}E}</annotation>

</semantics>

</math></span><img src="./81958fecc11482393d229957400a2c56d478d05a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.066ex; height:2.509ex;" alt="{\displaystyle A{\text{, }}B{\text{, and }}E}" loading="lazy"></span> are scalar position coefficients</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> is the <a href="Gravitational_parameter" class="mw-redirect" title="Gravitational parameter">gravitational parameter</a> of the focal body of the orbiting body</li></ul>

<p>Find the root of the scalar distance polynomial for the second observation of the orbiting body:

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {r_{2}}^{8}+a{r_{2}}^{6}+b{r_{2}}^{3}+c=0}</annotation>

</semantics>

</math></span></span>

where

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

</semantics>

</math></span><img src="./4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" loading="lazy"></span> is the scalar distance for the second observation of the orbiting body (it and its vector, <b>r<sub>2</sub></b>, are in the Equatorial Coordinate System)</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a{\text{, }}b{\text{, and }}c}</annotation>

</semantics>

<p>Various methods can be used to find the root, a suggested method is the <a href="Newton%E2%80%93Raphson_method" class="mw-redirect" title="Newton–Raphson method">Newton–Raphson method</a>. The root must be physically possible (i.e., not negative nor complex) and if multiple roots are suitable, each must be evaluated and compared to any available data to confirm their validity.

</p>

<p>Calculate the <a href="Slant_range" title="Slant range">slant range</a>, the distance from the observer point to the orbiting body at their respective time:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho _{1}&={\frac {1}{D_{0}}}\left[{\frac {6\left(D_{31}{\dfrac {\tau _{1}}{\tau _{3}}}+D_{21}{\dfrac {\tau }{\tau _{3}}}\right){r_{2}}^{3}+\mu D_{31}\left(\tau ^{2}-{\tau _{1}}^{2}\right){\dfrac {\tau _{1}}{\tau _{3}}}}{6{r_{2}}^{3}+\mu \left(\tau ^{2}-{\tau _{3}}^{2}\right)}}-D_{11}\right]\\\rho _{2}&=A+{\frac {\mu B}{{r_{2}}^{3}}}\\\rho _{3}&={\frac {1}{D_{0}}}\left[{\frac {6\left(D_{13}{\dfrac {\tau _{3}}{\tau _{1}}}-D_{23}{\dfrac {\tau }{\tau _{1}}}\right){r_{2}}^{3}+\mu D_{13}\left(\tau ^{2}-{\tau _{3}}^{2}\right){\dfrac {\tau _{3}}{\tau _{1}}}}{6{r_{2}}^{3}+\mu \left(\tau ^{2}-{\tau _{1}}^{2}\right)}}-D_{33}\right]\end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{n}}</annotation>

</semantics>

</math></span><img src="./aae86a9b5e3ef50889d51507f5f05a1dd66fd356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle \rho _{n}}" loading="lazy"></span> is the respective <a href="Slant_range" title="Slant range">slant range</a> (it and its vector, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\rho _{n}} }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\rho _{n}} }</annotation>

</semantics>

</math></span><img src="./f693380e9614adff06fccb251f3a27a1fe511dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.484ex; height:2.176ex;" alt="{\displaystyle \mathbf {\rho _{n}} }" loading="lazy"></span>, are in the Topocentric Equatorial Coordinate System)</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{mn}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{(n)}}</annotation>

</semantics>

</math></span><img src="./8090df9b47fa485b3ecec32c4d52b611bb6eb8f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.514ex; height:2.509ex;" alt="{\displaystyle \tau _{(n)}}" loading="lazy"></span> is the time interval.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

<p>Calculate the orbiting body position vectors, by adding the observer position vector to the slant direction vector (which is the slant distance multiplied by the slant direction vector):

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {r_{1}} &=\mathbf {R_{1}} +\rho _{1}\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \\[1.7ex]\mathbf {r_{2}} &=\mathbf {R_{2}} +\rho _{2}\mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \\[1.7ex]\mathbf {r_{3}} &=\mathbf {R_{3}} +\rho _{3}\mathbf {{\hat {\boldsymbol {\rho }}}_{3}} \end{aligned}}}">

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r_{1}} &=\mathbf {R_{1}} +\rho _{1}\mathbf {{\hat {\boldsymbol {\rho }}}_{1}} \\[1.7ex]\mathbf {r_{2}} &=\mathbf {R_{2}} +\rho _{2}\mathbf {{\hat {\boldsymbol {\rho }}}_{2}} \\[1.7ex]\mathbf {r_{3}} &=\mathbf {R_{3}} +\rho _{3}\mathbf {{\hat {\boldsymbol {\rho }}}_{3}} \end{aligned}}}</annotation>

</semantics>

</math></span></span>

where

</p>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r_{n}} }</annotation>

</semantics>

</math></span><img src="./7bc1ffdf7dbf80929c0938f44118e3aab5a777d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.009ex;" alt="{\displaystyle \mathbf {r_{n}} }" loading="lazy"></span> is the respective orbiting body position vector (in <a href="Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial Coordinate System</a>)</li>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{n}} }</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{n}}</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {{\hat {\boldsymbol {\rho }}}_{n}} }</annotation>

</semantics>

<p>Calculate the Lagrange coefficients:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{1}&\approx 1-{\frac {1}{2}}{\frac {\mu }{{r_{2}}^{3}}}{\tau _{1}}^{2}\\f_{3}&\approx 1-{\frac {1}{2}}{\frac {\mu }{{r_{2}}^{3}}}{\tau _{3}}^{2}\\g_{1}&\approx \tau _{1}-{\frac {1}{6}}{\frac {\mu }{{r_{2}}^{3}}}{\tau _{1}}^{3}\\g_{3}&\approx \tau _{3}-{\frac {1}{6}}{\frac {\mu }{{r_{2}}^{3}}}{\tau _{3}}^{3}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

where,

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}}</annotation>

</semantics>

</math></span><img src="./50dfd257a51e037112c917f8a9e47c9c053466df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{1}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{3}}</annotation>

</semantics>

</math></span><img src="./8e542b30556f75f837f72ea05a3940ddf38937d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{3}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math></span><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

</semantics>

</math></span><img src="./207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"></span> are the <a href="Lagrange_polynomial" title="Lagrange polynomial">Lagrange coefficients</a> (these are just the first two terms of the series expression based on the assumption of small time interval)</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{(n)}}</annotation>

</semantics>

<p>Calculate the velocity vector for the second observation of the orbiting body:

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v_{2}} ={\frac {1}{f_{1}g_{3}-f_{3}g_{1}}}\left(-f_{3}\mathbf {r_{1}} +f_{1}\mathbf {r_{3}} \right)}</annotation>

</semantics>

</math></span></span>

where

</p>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v_{2}} }</annotation>

</semantics>

</math></span><img src="./d7514b46fb3ba49c8335b11c55c5f10e9d2bfbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.588ex; height:2.009ex;" alt="{\displaystyle \mathbf {v_{2}} }" loading="lazy"></span> is the velocity vector for the second observation of the orbiting body (in <a href="Equatorial_coordinate_system" title="Equatorial coordinate system">Equatorial Coordinate System</a>)</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{3}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r_{n}} }</annotation>

</semantics>

<p>The <a href="Orbital_state_vectors" title="Orbital state vectors">orbital state vectors</a> have now been found, the position (<b>r</b><sub>2</sub>) and velocity (<b>v</b><sub>2</sub>) vector for the second observation of the orbiting body. With these two vectors, the <a href="Orbital_elements" title="Orbital elements">orbital elements</a> can be found and the orbit determined.

</p>

<ul><li><a href="Ellipse#Ellipses" title="Ellipse">Inscribed angle theorem and three-point form for ellipses</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Curtis, Howard D. <i><a href="Orbital_Mechanics_for_Engineering_Students" title="Orbital Mechanics for Engineering Students">Orbital Mechanics for Engineering Students</a></i>. Oxford: Elsevier Butterworth-Heinemann, 2005. Print.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://copernico.dm.unipi.it/~gronchi/PDF/gronchi_discuss.pdf">Gronchi, Giovanni F.. "Classical and modern orbit determination for asteroids." Proceedings of the International Astronomical Union2004.IAUC196 (2004): 1-11. Print.</a></span>

</li>

</ol></div></div>

<ul><li>Der, Gim J.. "New Angles-only Algorithms for Initial Orbit Determination." Advanced Maui Optical and Space Surveillance Technologies Conference. (2012). Print.</li></ul>

<p><br>

</p>

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</style><div id="Carl_Friedrich_Gauss153" style="font-size:114%;margin:0 4em"><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Gauss_composition_law" title="Gauss composition law">Gauss composition law</a></li>

<li><a href="Gauss_map" title="Gauss map">Gauss map</a></li>

<li><a href="Gauss_notation" title="Gauss notation">Gauss notation</a></li>

<li><a href="Gaussian_brackets" title="Gaussian brackets">Gaussian brackets</a></li>

<li><a href="Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a></li>

<li><a href="Gaussian_period" title="Gaussian period">Gaussian period</a></li>

<li><a href="Gaussian_surface" title="Gaussian surface">Gaussian surface</a></li>

<li><a href="Gaussian_units" title="Gaussian units">Gaussian units</a></li>

<li><a href="Gauss's_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a></li>

<li><a href="Gauss's_law" title="Gauss's law">Gauss's law</a></li>

<li><a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></li></ul>

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