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<html> <head> <title> Physics - Celestial Mechanics </title> </head> <body bgcolor="ffffff" text="000000" link="0000ff" alink="ff0000" vlink="00007f">  <center><h3>Physics topics</h3><br>by Dr. J. B. Tatum<br><a href="mailto:tatumjb352@gmail.com">tatumjb352@gmail.com</a></center><br><hr size="5" width="80%" color="007FFF"><br> <table width="100%" border="0"> <tr valign="top"> <td width="10%"> <a href="index.html">Home</a><br><br> <a href="stellatm.html">Stellar Atmospheres</a><br><br> <a href="celmechs.html">Celestial Mechanics</a><br><br> <a href="classmechs.html">Classical Mechanics</a><br><br> <a href="goptics.html">Geometric Optics</a><br><br> <a href="elmag.html">Electricity and Magnetism</a><br><br> <a href="thermod.html">Heat and Thermodynamics</a><br><br> <a href="physopt.html">Physical Optics</a><br><br> <a href="plphot.html">Max Fairbairn's Planetary Photometry</a><br><br> <a href="integrals.html">Integrals and Differential Equations</a><br><br> <a href="quadric_surfaces.html">Quadric Surfaces</a><br><br> </td> <td width="2%"></td>  <td>  <center><b><u>Celestial Mechanics</u></b> (last <a href="celmechs/update.html">updated</a>: 2023 January 16)</center><br><br> <font face="Times New Roman"> <b>Part I. Mathematical Preambles</b><br><br> <a href="celmechs/celm1.pdf">Chapter 1.</a>    Numerical Methods<br><br> <table border="0"> <tr><td>1.1</td><td>Introduction</td></tr> <tr><td>1.2</td><td>Numerical Integration</td></tr> <tr><td>1.3</td><td>Quadratic Equations</td></tr> <tr><td>1.4</td><td>The Solution of <i>f</i>(<i>x</i>) = 0</td></tr> <tr><td>1.5</td><td>The Solution of Polynomial Equations</td></tr> <tr><td>1.6</td><td>Failure of the Newton-Raphson Method</td></tr> <tr><td>1.7</td><td>Simultaneous Linear Equations, <i>N</i> = <i>n</i></td></tr> <tr><td>1.8</td><td>Simultaneous Linear Equations, <i>N</i> < <i>n</i></td></tr> <tr><td>1.9</td><td>Nonlinear Simultaneous Equations</td></tr> <tr><td>1.10</td><td>Besselian Interpolation</td></tr> <tr><td>1.11</td><td>Fitting a Polynomial to a Set of Points. Lagrange Polynomials. Lagrange Interpolation.</td></tr> <tr><td>1.12</td><td>Fitting a Least Squares Straight Line to a Set of Observational Points</td></tr> <tr><td>1.13</td><td>Fitting a Least Squares Polynomial to a Set of Observational Points</td></tr> <tr><td>1.14</td><td>Legendre Polynomials</td></tr> <tr><td>1.15</td><td>Gaussian Quadrature - the Algorithm</td></tr> <tr><td>1.16</td><td>Gaussian Quadrature - Derivation</td></tr> <tr><td>1.17</td><td>Frequently-needed Numerical Procedures</td></tr> </table><br><br> <a href="celmechs/celm2.pdf">Chapter 2.</a>    Conic Sections<br><br> <table border="0"> <tr><td>2.1</td><td>Introduction</td></tr> <tr><td>2.2</td><td>The Straight Line</td></tr> <tr><td>2.3</td><td>The Ellipse</td></tr> <tr><td>2.4</td><td>The Parabola</td></tr> <tr><td>2.5</td><td>The Hyperbola</td></tr> <tr><td>2.6</td><td>Conic Sections</td></tr> <tr><td>2.7</td><td>The General Conic Section</td></tr> <tr><td>2.8</td><td>Fitting a Conic Section Through Five Points</td></tr> <tr><td>2.9</td><td>Fitting a Conic Section Through <i>n</i> Points</td></tr> </table><br><br> <a href="celmechs/celm3.pdf">Chapter 3.</a>    Plane and Spherical Trigonomtry</br></br> <table border="0"> <tr><td>3.1</td><td>Introduction</td></tr> <tr><td>3.2</td><td>Plane Triangles</td></tr> <tr><td>3.3</td><td>Cylindrical and Spherical Coordinates</td></tr> <tr><td>3.4</td><td>Velocity and Acceleration Components</td></tr> <tr><td>3.5</td><td>Spherical Triangles</td></tr> <tr><td>3.6</td><td>Rotation of Axes, Two Dimensions</td></tr> <tr><td>3.7</td><td>Rotation of Axes, Three Dimensions. Eulerian Angles</td></tr> <tr><td>3.8</td><td>Trigonometrical Formulas</td></tr> </table><br><br> <a href="celmechs/celm4.pdf">Chapter 4.</a>    Coordinate Geometry in Three Dimensions<br><br> <table border="0"> <tr><td>4.1</td><td>Introduction</td></tr> <tr><td>4.2</td><td>Planes and Straight Lines</td></tr> <tr><td>4.3</td><td>The Ellipsoid</td></tr> <tr><td>4.4</td><td>The Paraboloid</td></tr> <tr><td>4.5</td><td>The Hyperboloid</td></tr> <tr><td>4.6</td><td>The Cylinder</td></tr> <tr><td>4.7</td><td>The Cone</td></tr> <tr><td>4.8</td><td>The General Second Degree Equation in Three Dimensions</td></tr> <tr><td>4.9</td><td>Matrices</td></tr> </table><br><br> <a href="celmechs/celm5.pdf">Chapter 5.</a>    Gravitational Field and Potential<br><br> <table border="0"> <tr><td>5.1</td><td>Introduction</td></tr> <tr><td>5.2</td><td>Gravitational Field</td></tr> <tr><td>5.3</td><td>Newton's Law of Gravitation</td></tr> <tr><td>5.4</td><td>The Gravitational Fields Around Various Bodies</td></tr> <tr><td></td><td> <table border="0"> <tr><td>5.4.1</td><td>Gravitational Field Around a Point Mass</td></tr> <tr><td>5.4.2</td><td>Gravitational Field on the Axis of a Ring</td></tr> <tr><td>5.4.3</td><td>Plane discs</td></tr> <tr><td>5.4.4</td><td>Infinite Plane Laminas</td></tr> <tr><td>5.4.5</td><td>Rods</td></tr> <tr><td>5.4.6</td><td>Solid Cylinder</td></tr> <tr><td>5.4.7</td><td>Hollow Spherical Shell</td></tr> <tr><td>5.4.8</td><td>Uniform Solid Sphere</td></tr> <tr><td>5.4.9</td><td>Bubble Inside a Uniform Solid Sphere</td></tr> <tr><td>5.4.10</td><td>Field Inside a Nonuniform Sphere</td></tr> <tr><td></td><td> <table border="0"> <tr><td>5.4.10.A</td><td>Differentiated Planet</td></tr> <tr><td>5.4.10.B</td><td>Undifferentiated Planet</td></tr> <tr><td></td><td> <table border="0"> <tr><td>5.4.10.B.i</td><td>Density Increases Uniformly With Depth</td></tr> <tr><td>5.4.10.B.i</td><td>Density Increases Nonuniformly With Depth</td></tr> </table> </td></tr> </table> </td></tr> </table> </td></tr> <tr><td>5.5</td><td>Gauss's Theorem</td></tr> <tr><td>5.6</td><td>Calculating Surface Integrals</td></tr> <tr><td>5.7</td><td>Pressure at the Centre of a Uniform Solid Sphere</td></tr> <tr><td>5.8</td><td>Gravitational Potential</td></tr> <tr><td>5.9</td><td>Nabla, Gradient and Divergence. Poisson's and Laplace's Equations</td></tr> <tr><td>5.10</td><td>The Gravitational Potential Near Various Bodies</td></tr> <tr><td></td><td> <table border="0"> <tr><td>5.10.1</td><td>Potential Near a Point Mass</td></tr> <tr><td>5.10.2</td><td>Potential on the Axis of a Ring</td></tr> <tr><td>5.10.3</td><td>Plane Discs</td></tr> <tr><td>5.10.4</td><td>Infinite Plane Lamina</td></tr> <tr><td>5.10.5</td><td>Hollow Hemisphere</td></tr> <tr><td>5.10.6</td><td>Rods</td></tr> <tr><td>5.10.7</td><td>Solid Cylinder</td></tr> <tr><td>5.10.8</td><td>Hollow Spherical Shell</td></tr> <tr><td>5.10.9</td><td>Uniform Solid Sphere</td></tr> <tr><td></td><td> <table border="0"> <tr><td>5.10.9.A</td><td>Potential Inside and Outside</td></tr> <tr><td>5.10.9.B</td><td>Work Required to Assemble a Uniform Sphere</td></tr> </table> </td></tr> </table> </td></tr> <tr><td>5.11</td><td>Legendre Polynomials</td></tr> <tr><td>5.12</td><td>Gravitational Potential in the Vicinity of any Massive Body</td></tr> </table><br><br> <b>Part II. Celestial Mechanics</b><br><br> <a href="celmechs/celm6.pdf">Chapter 6.</a>    The Celestial Sphere<br><br> <table border="0"> <tr><td>6.1</td><td>Introduction</td></tr> <tr><td>6.2</td><td>Altazimuth Coordinates</td></tr> <tr><td>6.3</td><td>Equatorial Coordinates</td></tr> <tr><td>6.4</td><td>Conversion Between Equatorial and Altazimuth Coordinates</td></tr> <tr><td>6.5</td><td>Ecliptic Coordinates</td></tr> <tr><td>6.6</td><td>The Mean Sun</td></tr> <tr><td>6.7</td><td>Precession</td></tr> <tr><td>6.8</td><td>Nutation</td></tr> <tr><td>6.9</td><td>The Length of the Year</td></tr> </table><br><br> <a href="celmechs/celm7.pdf">Chapter 7.</a>    Time<br><br> <a href="celmechs/celm8.pdf">Chapter 8.</a>    Planetary Motions<br><br> <table border="0"> <tr><td>8.1</td><td>Introduction</td><tr> <tr><td>8.2</td><td>Opposition, Conjunction and Quadrature</td><tr> <tr><td>8.3</td><td>Sidereal and Synodic Periods</td><tr> <tr><td>8.4</td><td>Direct and Retrograde Motion, and Stationary Points</td><tr> </table><br><br> <a href="celmechs/celm9.pdf">Chapter 9.</a>    The Two Body Problem in Two Dimensions<br><br> <table border="0"> <tr><td>9.1</td><td>Introduction</td></tr> <tr><td>9.2</td><td>Kepler's Laws</td></tr> <tr><td>9.3</td><td>Kepler's Second Law from Conservation of Angular Momentum</td></tr> <tr><td>9.4</td><td>Some Functions of the Masses</td></tr> <tr><td>9.5</td><td>Kepler's First and Third Laws from Newton's Law of Gravitation</td></tr> <tr><td>9.6</td><td>Position in an Elliptic Orbit</td></tr> <tr><td>9.7</td><td>Position in a Parabolic Orbit</td></tr> <tr><td>9.8</td><td>Position in a Hyperbolic Orbit</td></tr> <tr><td>9.9</td><td>Orbital Elements and Velocity Vector</td></tr> <tr><td>9.10</td><td>Osculating Elements</td></tr> <tr><td>9.11</td><td>Mean Distance in an Elliptic Orbit</td></tr> </table><br><br> <a href="celmechs/celm10.pdf">Chapter 10.</a>    Computation of an Ephemeris<br><br> <table border="0"> <tr><td>10.1</td><td>Introduction</td></tr> <tr><td>10.2</td><td>Elements of an Elliptic Orbit</td></tr> <tr><td>10.3</td><td>Some Additional Angles</td></tr> <tr><td>10.4</td><td>Elements of a Circular or Near-circular Orbit</td></tr> <tr><td>10.5</td><td>Elements of a Parabolic Orbit</td></tr> <tr><td>10.6</td><td>Elements of a Hyperbolic Orbit</td></tr> <tr><td>10.7</td><td>Calculating the Position of a Comet or Asteroid</td></tr> <tr><td>10.8</td><td>Quadrant Problems</td></tr> <tr><td>10.9</td><td>Computing an Ephemeris</td></tr> <tr><td>10.10</td><td>Orbital Elements and Velocity Vector</td></tr> <tr><td>10.11</td><td>Hamiltonian Formulation of the Equations of Motion</td></tr> </table><br><br> <a href="celmechs/celm11.pdf">Chapter 11.</a>    Photographic Astrometry<br><br> <table border="0"> <tr><td>11.1</td><td>Introduction</td></tr> <tr><td>11.2</td><td>Standard Coordinates and Plate Constants</td></tr> <tr><td>11.3</td><td>Refinements and Corrections</td></tr> <td></td><td> <table border="0"> <tr><td>11.3.1</td><td>Parallaxes of the Comparison Stars</td></tr> <tr><td>11.3.2</td><td>Proper Motions of the Comparison Stars</td></tr> <tr><td>11.3.3</td><td>Refraction</td></tr> <tr><td>11.3.4</td><td>Aberration of light</td></tr> <tr><td>11.3.5</td><td>Optical Distortion</td></tr> <tr><td>11.3.6</td><td>Errors, Mistakes and Blunders</td></tr> </table> </td></tr> </table><br><br> <a href="celmechs/celm12.pdf">Chapter 12.</a>    CCD Astrometry<br><br> <a href="celmechs/celm13.pdf">Chapter 13.</a>    Calculation of Orbital Elements<br><br> <table border="0"> <tr><td>13.1</td><td>Introduction</td></tr> <tr><td>13.2</td><td>Triangles</td></tr> <tr><td>13.3</td><td>Sectors</td></tr> <tr><td>13.4</td><td>Kepler's Second Law</td></tr> <tr><td>13.5</td><td>Coordinates</td></tr> <tr><td>13.6</td><td>Example</td></tr> <tr><td>13.7</td><td>Geocentric and Heliocentric Distances - First Attempt</td></tr> <tr><td>13.8</td><td>Improved Triangle Ratios</td></tr> <tr><td>13.9</td><td>Iterating</td></tr> <tr><td>13.10</td><td>Higher-order Approximation</td></tr> <tr><td>13.11</td><td>Light-time Correction</td></tr> <tr><td>13.12</td><td>Sector-Triangle Ratio</td></tr> <tr><td>13.13</td><td>Resuming the Numerical Example</td></tr> </table><br><br> <a href="celmechs/celm14.pdf">Chapter 14.</a>    General Perturbation Theory<br><br> <table border="0"> <tr><td>14.1</td><td>Introduction</td></tr> <tr><td>14.2</td><td>Contact Transformations and General Perturbation Theory</td></tr> <tr><td>14.3</td><td>The Poisson Brackets for the Orbital Elements</td></tr> <tr><td>14.4</td><td>Lagrange's Planetary Equations</td></tr> <tr><td>14.5</td><td>Motion Around an Oblate Symmetric Top</td></tr> </table><br><br> <a href="celmechs/celm15.pdf">Chapter 15.</a>    Special Perturbations<br><br> <table border="0"> <tr><td>15.1</td><td>Introduction</td></tr> <tr><td>15.2</td><td>Orbital Elements and the Position and Velocity Vector</td></tr> <tr><td>15.3</td><td>The Equations of Motion</td></tr> </table><br><br> <a href="celmechs/celm16.pdf">Chapter 16.</a>    Equivalent Potential and the Restricted Three-Body Problem<br><br> <table border="0"> <tr><td>16.1</td><td>Introduction</td></tr> <tr><td>16.2</td><td>Motion Under a Central Force</td></tr> <tr><td>16.3</td><td>Inverse Square Attractive Force</td></tr> <tr><td>16.4</td><td>Hooke's Law</td></tr> <tr><td>16.5</td><td>Inverse Fourth Power Force</td></tr> <tr><td>16.6</td><td>The Collinear Lagrangian Points</td></tr> <tr><td>16.7</td><td>The Equilateral Lagrangian Points</td></tr> </table><br><br> <a href="celmechs/celm17.pdf">Chapter 17.</a>    Visual Binary Stars<br><br> <table border="0"> <tr><td>17.1</td><td>Introduction</td></tr> <tr><td>17.2</td><td>Determination of the Apparent Orbit</td></tr> <tr><td>17.3</td><td>The Elements of the True Orbit</td></tr> <tr><td>17.4</td><td>Determination of the Elements of the True Orbit</td></tr> <tr><td>17.5</td><td>Construction of an Ephemeris</td></tr> </table><br><br> <a href="celmechs/celm18.pdf">Chapter 18.</a>    Spectroscopic Binary Stars<br><br> <table border="0"> <tr><td>18.1</td><td>Introduction</td></tr> <tr><td>18.2</td><td>The Velocity Curve from the Elements</td></tr> <tr><td>18.3</td><td>Preliminary Elements from the Velocity Curve</td></tr> <tr><td>18.4</td><td>Masses</td></tr> <tr><td>18.5</td><td>Refinement of the Orbital Elements</td></tr> <tr><td>18.6</td><td>Finding the Period</td></tr> <tr><td>18.7</td><td>Measuring the Radial Velocity</td></tr> </table><br><br> <a href="celmechs/celmA.pdf">Appendix A.</a>    Miscellaneous Problems<br><br> <a href="celmechs/celmB.pdf">Appendix B.</a>    Solutions to Miscellaneous Problems<br><br> </font> </table> <br> <hr size="5" width="80%" color="007FFF"><br> <font size="1"> Texts © 2000 - 2013 Dr. J. B. Tatum<br> Web page design and code © 2002 - 2013 Jason Stumpf </font> </body> </html>