13: Calculation of Orbital Elements
- Page ID
- 6874
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 13.1: Introduction to Calculating Orbital Elements
- This page discusses the complexities of determining orbital elements from observational data, stressing the need for three observations to compute six elements without knowing the planet's distance. It emphasizes the importance of correcting for observational errors and distinguishing them from mistakes, and addresses challenges like time discrepancies and parallax corrections.
- 13.2: Triangles
- This page explores a geometric theorem related to triangles and coplanar vectors, emphasizing linear combinations. It illustrates how to express one vector, \(\textbf{r}_2\), as a combination of two others, \(\textbf{r}_1\) and \(\textbf{r}_3\), using coefficients derived from the areas of triangles formed by the vectors. The process involves vector cross products to determine these coefficients and establish a connection between the areas of triangles and the vectors.
- 13.3: Sectors
- This page covers conic section orbits, emphasizing sector areas formed by radius vectors and their relationships through triangle ratios. It relates these area ratios to time intervals according to Kepler's second law, presenting equations for better understanding. The goal is to use these approximations for determining heliocentric distances at certain observations, with a preview of further topics to be explored in later sections.
- 13.4: Kepler's Second Law
- This page covers Kepler's second law, highlighting the equal area property in two-body systems and the significance of celestial body masses compared to the Sun. It introduces the gravitational constant \(G\) and the mass of the Sun, while explaining the redefinition of the astronomical unit by the International Astronomical Union in 2012. The section concludes with a discussion on the relationship between gravitational force, centripetal acceleration, and the astronomical unit.
- 13.5: Coordinates
- This page provides an overview of coordinate systems in celestial mechanics, detailing heliocentric systems like plane-of-orbit, ecliptic, and equatorial coordinates, each with specific axes and measurements. It also introduces the geocentric system, emphasizing angular measurements relative to Earth. Additionally, it includes equations that connect geocentric coordinates to direction cosines, showcasing the relationships between these various coordinate frameworks.
- 13.6: Example
- This page presents a numerical example using observational data from the minor planet 2 Pallas, detailing the context of the observations made from the Earth's center at a specific time. It stresses the significance of actual observations reported in Universal Time and from the Earth's surface, and addresses potential discrepancies caused by rounding and limited data.
- 13.7: Geocentric and Heliocentric Distances - First Attempt
- This page focuses on heliocentric equatorial components relevant to celestial mechanics, introducing equations for heliocentric coordinates and methods for calculating geocentric distances. It includes numerical examples, verification through direction cosines, and comparisons with MPC ephemeris values, highlighting the need for refinement in calculations.
- 13.8: Improved Triangle Ratios
- This page covers the motion of orbiting bodies, detailing equations of motion in heliocentric coordinates, deriving acceleration components, and introducing higher derivatives via Taylor expansion. It also examines angular momentum and areal speed, relating them to area swept in orbital motion, which precedes Kepler's laws.
- 13.9: Iterating
- This page explores enhancing triangle ratio estimates through specific equations and numerical data, presenting initial estimates and iterative results for values \(a_1\) and \(a_3\). It notes significant improvements from iterations while highlighting that later updates provide minor changes.
- 13.10: Higher-order Approximation
- This page focuses on calculating the radial velocity of a particle in solar orbit, outlining formulas for different orbital types (elliptical, parabolic, hyperbolic) that require knowing the orbital eccentricity. It explains how to determine heliocentric distances at specific times to form a quadratic expression for radial velocity. The page also covers further calculations for enhanced accuracy using derived equations for additional terms, culminating in the values of \(a_1\) and \(a_3\).
- 13.11: Light-time Correction
- This page emphasizes the significance of light-time corrections for accurately determining planetary positions from Earth, noting discrepancies between observed and actual positions due to light's finite speed. It suggests that correcting these observed times can yield more precise geocentric distances, and while the recalculation process may appear complex, it can be automated with computers.
- 13.12: Sector-Triangle Ratio
- This page covers the determination of area ratios for sectors and triangles in planetary motion, linked to Kepler's laws, including mathematical relationships to calculate sector-triangle ratios \(R_1\), \(R_2\), and \(R_3\). It presents methods for eliminating variables from equations in celestial mechanics, aiming to solve for \(R_3\) and other constants.
- 13.13: Resuming the Numerical Example
- This page covers the iterative refinement of celestial coordinates and angles for the minor planet Pallas through numerical methods. It details the calculation of the angle \(f_3\) using heliocentric coordinates and explores the true anomaly, emphasizing the object's eccentric orbit.
- 13.14: Summary So Far
- This page details a method for calculating geocentric and heliocentric distances of a planet using observational data. It presents a structured approach that involves time conversion, solar coordinate calculations, and iterative distance refinements. Key considerations include correcting for light travel time and accurately calculating angles with trigonometric functions, particularly the recommended \(\text{ATAN2}\) function to avoid quadrant errors.
- 13.15: Calculating the Elementss
- This page covers the computation of the semi latus rectum and eccentricity of an ellipse using polar equations, resulting in consistent orbital values. It details the calculation of the semi-major axis, orbital period, time of perihelion passage, and direction cosines, leading to the determination of key angles. The inclination is calculated at approximately 35.21 degrees, and a summary of orbital elements is provided.
- 13.16: Topocentric-Geocentric Correction
- This page covers essential corrections for precise asteroid observations, focusing on geocentric and topocentric coordinates. It details the \(∆T\) correction and light-time correction, alongside the topocentric-geocentric adjustment due to Earth's diameter. Formulas for converting topocentric coordinates to geocentric ones are provided, emphasizing the role of Observatory Codes in tracking specific observation parameters.
- 13.17: Concluding Remarks
- This page provides foundational skills for determining elliptical orbit elements from three observations and encourages further study for advanced concepts like multiple observations and non-elliptical orbits. It highlights the challenges of starting orbital calculations and aims to simplify this process for readers, while indicating that more complex topics will require additional learning and experience.
Thumbnail: Illustration of Kepler's second law of planetary orbits. Illustration by RJHall