10: Computation of an Ephemeris
- Page ID
- 6853
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.1: Introduction to an Ephemeris
- The entire enterprise of determining the orbits of planets, asteroids and comets is quite a large one, involving several stages. From the available observations, the orbit of the body has to be determined; in particular we have to determine the orbital elements, a set of parameters that describe the orbit. For a new body, one determines preliminary elements from the initial few observations that have been obtained. As more observations are accumulated, so will the calculated preliminary elements
- 10.2: Elements of an Elliptic Orbit
- This page describes the six elements defining an elliptic orbit in three dimensions: semi-major axis (\(a\)), eccentricity (\(e\)), inclination (\(i\)), longitude of the ascending node (\(Ω\)), argument of perihelion (\(ω\)), and time of perihelion passage (\(T\)). It emphasizes that \(i\), \(Ω\), and \(ω\) are referenced to the equinox and equator of a specified epoch, typically J2000.0, detailing the geometric significance of inclination and longitude of the ascending node.
- 10.3: Some Additional Angles
- This page covers key angles in planetary motion, emphasizing true anomaly, mean anomaly, and angular measurements. It defines the sum of angles Ω and ω as ϖ, the longitude of perihelion, and describes true anomaly \(v\) as the planet's position. The mean anomaly \(M\) represents a constant motion of a hypothetical mean planet, while the true longitude \(l\), despite its name, combines these angles across two planes. Mean longitude \(L\) similarly describes the mean planet's motion.
- 10.4: Elements of a Circular or Near-circular Orbit
- This page discusses the uncertainties in perihelion position and passage time for near-circular orbits, noting that these are undefined in perfect circular orbits. It explains the use of the argument of perihelion \(ω\) as a reference, while emphasizing the adoption of an epoch \(t_0\) and other parameters like mean anomaly \(M_0\) for clarity. Additionally, it highlights the importance of distinguishing this epoch from equinox and equator references in relation to angular elements.
- 10.5: Elements of a Parabolic Orbit
- The eccentricity, of course, is unity, so only five elements are necessary. In place of the semi major axis, one usually specifies the parabola by the perihelion distance q . Presumably no orbit is ever exactly parabolic, which implies an eccentricity of exactly one. However, many long-distance comets move in large and eccentric orbits, and we see them over such a short arc near to perihelion that it is not possible to calculate accurate elliptic orbits, and we usually then fit a parabolic orbi
- 10.6: Elements of a Hyperbolic Orbit
- This page explains the semi transverse axis \(a\) in hyperbolas, highlighting it as a nomenclature issue, with some authors treating \(a\) as negative in orbital calculations. It notes that, although interstellar comets could theoretically enter the solar system in hyperbolic orbits, none have been observed doing so. Instead, some comets may transition from elliptical to hyperbolic orbits influenced by Jupiter's gravity, leading to complex orbital dynamics that challenge long-term predictions.
- 10.7: Calculating the Position of a Comet or Asteroid
- We suppose that we are given the orbital elements of an asteroid or comet. Our task is to be able to predict, from these, the right ascension and declination of the object in the sky at some specified future (or past) date. If we can do it for one date, we can do it for many dates - e.g. every day for a year if need be. In other words, we will have constructed an ephemeris.