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- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/09%3A_The_Two_Body_Problem_in_Two_Dimensions/9.10%3A_Mean_Distance_in_an_Elliptic_OrbitIt is sometimes said that “ a ” in an elliptic orbit is the “mean distance” of a planet from the Sun. In fact a is the semi major axis of the orbit. Whether and it what sense it might also be the “m...It is sometimes said that “ a ” in an elliptic orbit is the “mean distance” of a planet from the Sun. In fact a is the semi major axis of the orbit. Whether and it what sense it might also be the “mean distance” is worth a moment of thought.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.04%3A_The_Gravitational_Fields_of_Various_Bodies/5.4.10%3A_Bubble_Inside_a_Uniform_Solid_SphereThe field at P is equal to the field due to the entire sphere minus the field due to the missing mass of the bubble. \[\textbf{g} = -\frac{4}{3} \pi G ρ \textbf{r}_1 - (-\frac{4}{3} \pi G ρ...The field at P is equal to the field due to the entire sphere minus the field due to the missing mass of the bubble. g=−43πGρr1−(−43πGρr2)=−43πGρ(r1−r2)=−43πGρc. is independent of the position of P) and is parallel to the line joining the centres of the two spheres.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.08%3A_Fitting_a_Conic_Section_Through_n_PointsThe five normal Equations can then be set up and solved to give those values for the coefficients that will result in the sum of the squares of the residuals being least, and it is in that sense that ...The five normal Equations can then be set up and solved to give those values for the coefficients that will result in the sum of the squares of the residuals being least, and it is in that sense that the "best" ellipse results. After all, suppose that we have several positions of a planet in orbit around the Sun, or several positions of the secondary component of a visual binary star with respect to its primary component; we can now fit an ellipse through these positions.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.02%3A_Potential_on_the_Axis_of_a_RingThis is the same for all such elements around the circumference of the ring, and the total potential is just the scalar sum of the contributions from all the elements. Here −dψ/dz gives the z-...This is the same for all such elements around the circumference of the ring, and the total potential is just the scalar sum of the contributions from all the elements. Here −dψ/dz gives the z-component of the field, and the minus sign correctly indicates that the field is directed in the negative z-direction.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/09%3A_The_Two_Body_Problem_in_Two_Dimensions/9.07%3A_Position_in_a_Hyperbolic_OrbitIf an interstellar comet were to encounter the solar system from interstellar space, it would pursue a hyperbolic orbit around the Sun. To date, no such comet with an original hyperbolic orbit has bee...If an interstellar comet were to encounter the solar system from interstellar space, it would pursue a hyperbolic orbit around the Sun. To date, no such comet with an original hyperbolic orbit has been found, although there is no particular reason why we might not find one some night. However, a comet with a near-parabolic orbit from the Oort belt may approach Jupiter on its way in to the inner solar system, and its orbit may be perturbed into a hyperbolic orbit.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/01%3A_Numerical_Methods/1.05%3A_The_Solution_of_Polynomial_EquationsThe Newton-Raphson method is very suitable for the solution of polynomial equations.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/12%3A_CCD_AstrometryIn practice, a stellar image is spread out over several pixels in two dimensions, each of several pixels holding a certain number of photons. (Not literally photons, of course, but electron-hole pairs...In practice, a stellar image is spread out over several pixels in two dimensions, each of several pixels holding a certain number of photons. (Not literally photons, of course, but electron-hole pairs, each of which has been generated by a single photon.) The software reads the number of photons in each of the pixels over which the stellar image is distributed, it fits a statistical distribution function (such as a two-dimensional gaussian function) to the image, and calculates the "centre of g…
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/18%3A_Spectroscopic_Binary_Stars/18.04%3A_MassesAnd if, further, we have a reasonable idea of the mass M of the star (we know its spectral type and luminosity class from its spectrum, and we can suppose that it obeys the well-established relation b...And if, further, we have a reasonable idea of the mass M of the star (we know its spectral type and luminosity class from its spectrum, and we can suppose that it obeys the well-established relation between mass and luminosity of main-sequence stars), then we can determine m 3 sin 3 i and hence, of course m sini.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.08%3A_The_Gravitational_Potentials_Near_Various_Bodies/5.8.09%3A_Solid_SphereThe potential outside a solid sphere is just the same as if all the mass were concentrated at a point in the centre. We are going to find the potential at a point P inside a uniform sphere ...The potential outside a solid sphere is just the same as if all the mass were concentrated at a point in the centre. We are going to find the potential at a point P inside a uniform sphere of radius a, mass M, density ρ, at a distance r from the centre (r<a). The potential is in units of −GM/r, and distance is in units of a, the radius of the sphere.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.04%3A_The_Gravitational_Fields_of_Various_BodiesIn this section we calculate the fields near various shapes and sizes of bodies, much as one does in an introductory electricity course. Some of this will not have much direct application to celestial...In this section we calculate the fields near various shapes and sizes of bodies, much as one does in an introductory electricity course. Some of this will not have much direct application to celestial mechanics, but it will serve as good introductory practice in calculating fields and, later, potentials.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/01%3A_Numerical_Methods/1.06%3A_Failure_of_the_Newton-Raphson_MethodIn nearly all cases encountered in practice Newton-Raphson method is very rapid and does not require a particularly good first guess. Nevertheless for completeness it should be pointed out that there...In nearly all cases encountered in practice Newton-Raphson method is very rapid and does not require a particularly good first guess. Nevertheless for completeness it should be pointed out that there are rare occasions when the method either fails or converges rather slowly.
