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    About 176 results
    • https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/06%3A_The_Celestial_Sphere/6.07%3A_Precession
      From the point of view of classical mechanics, Earth is an oblate symmetric top. That is to say, it has an axis of symmetry and two of its principal moments of inertia are equal and are less than the ...From the point of view of classical mechanics, Earth is an oblate symmetric top. That is to say, it has an axis of symmetry and two of its principal moments of inertia are equal and are less than the moment of inertia about the axis of symmetry. The phenomena of precession of such a body are well understood and are studied in courses of classical mechanics. It is necessary, however, to be clear in one’s mind about the distinction between torque-free precession and torque-induced precession.
    • 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_Sphere
      The field at \(\text{P}\) 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 \(\text{P}\) is equal to the field due to the entire sphere minus the field due to the missing mass of the bubble. \[\textbf{g} = -\frac{4}{3} \pi G ρ \textbf{r}_1 - (-\frac{4}{3} \pi G ρ \textbf{r}_2) = -\frac{4}{3} \pi G ρ ( \textbf{r}_1 - \textbf{r}_2) = -\frac{4}{3} \pi G ρ \textbf{c}. \label{5.4.26} \tag{5.4.26}\] is independent of the position of \(\text{P}\)) and is parallel to the line joining the centres of the two spheres.
    • 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_Orbit
      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 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)/12%3A_CCD_Astrometry
      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...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_Masses
      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 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/5.04%3A_The_Gravitational_Fields_of_Various_Bodies
      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...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_Method
      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...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.
    • https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/17%3A_Visual_Binary_Stars
      A visual binary is a gravitationally bound system that can be resolved into two stars. These stars are estimated, via Kepler's 3rd law, to have periods ranging from a number of years to thousands of y...A visual binary is a gravitationally bound system that can be resolved into two stars. These stars are estimated, via Kepler's 3rd law, to have periods ranging from a number of years to thousands of years. A visual binary consists of two stars, usually of a different brightness.
    • https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/08%3A_Planetary_Motions
      In this chapter, I do not attempt to calculate planetary ephemerides, which will come in a later chapter. Rather, I discuss in an idealistic and qualitative manner how it is that a planet sometimes mo...In this chapter, I do not attempt to calculate planetary ephemerides, which will come in a later chapter. Rather, I discuss in an idealistic and qualitative manner how it is that a planet sometimes moves in one direction and sometimes in another. That the treatment in this chapter is both idealistic and qualitative by no means implies that it will be devoid of Equations or of quantitative results, or that the matter discussed in this chapter will have no real practical or observational value.
    • https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/01%3A_Numerical_Methods/1.12%3A_Fitting_a_Least_Squares_Straight_Line_to_a_Set_of_Observational_Points
      the line such that the sum of the squares of the vertical residuals is least is often called loosely the “least squares straight line”. Technically, it is the least squares linear regression of \(y\) ...the line such that the sum of the squares of the vertical residuals is least is often called loosely the “least squares straight line”. Technically, it is the least squares linear regression of \(y\) upon \(x\). We have \(N\) simultaneous linear Equations of this sort for the two unknowns \(a_1\) and \(a_0\), and, for the least squares regression of \(y\) upon \(x,\) we have to find the values of \(a_1\) and \(a_0\) such that the sum of the squares of the residuals is least.
    • https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.05%3A_Conic_Sections
      A plane section of a cone is either an ellipse, a parabola or a hyperbola, depending on whether the angle that the plane makes with the base of the cone is less than, equal to or greater than the angl...A plane section of a cone is either an ellipse, a parabola or a hyperbola, depending on whether the angle that the plane makes with the base of the cone is less than, equal to or greater than the angle that the generator of the cone makes with its base. However, given the definitions of the ellipse, parabola and hyperbola that we have given, proof is required that they are in fact conic sections.

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