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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_(Fowler)/14%3A_Mathematics_for_Orbits/14.02%3A_The_EllipseThe simplest nontrivial planetary orbit is a circle. An ellipse is a circle scaled (squashed) in one direction.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Astronomy_1e_(OpenStax)/03%3A_Orbits_and_Gravity/3.01%3A_The_Laws_of_Planetary_MotionTycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of pl...Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period (P) and the semimajor axis (a) of an orbit is given by \(P^2 = a^3\) (when a is in units
- https://phys.libretexts.org/Courses/Grossmont_College/ASTR_110%3A_Astronomy_(Fitzgerald)/02%3A_History_of_Astronomy/2.04%3A_The_Laws_of_Planetary_MotionTycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of pl...Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period (P) and the semimajor axis (a) of an orbit is given by \(P^2 = a^3\)
- https://phys.libretexts.org/Courses/Merrimack_College/Conservation_Laws_Newton's_Laws_and_Kinematics_version_2.0/08%3A_C8)_Conservation_of_Energy-_Kinetic_and_Gravitational/8.03%3A_Universal_GravityThe situation we want to understand is the gravitational interaction near the Earth - in fact, very near the Earth, so that we can write the height of the object from the surface \(h\) is much smaller...The situation we want to understand is the gravitational interaction near the Earth - in fact, very near the Earth, so that we can write the height of the object from the surface \(h\) is much smaller than the radius of the Earth, \(h<<R_E\).
- https://phys.libretexts.org/Courses/Gettysburg_College/Gettysburg_College_Physics_for_Physics_Majors/08%3A_C8)_Conservation_of_Energy-_Kinetic_and_Gravitational/8.03%3A_The_Inverse-Square_LawHere I have put a subscript \(E\) on \(g\) to emphasize that this is the acceleration of gravity near the surface of the Earth, and that the same formula could be used to find the acceleration of grav...Here I have put a subscript \(E\) on \(g\) to emphasize that this is the acceleration of gravity near the surface of the Earth, and that the same formula could be used to find the acceleration of gravity near the surface of any other planet or moon, just replacing \(M_E\) and \(R_E\) by the mass and radius of the planet or moon in question.
- https://phys.libretexts.org/Courses/Merrimack_College/Conservation_Laws_Newton's_Laws_and_Kinematics_version_2.0/13%3A_Application_-_Orbits_and_Kepler's_Laws/13.01%3A_OrbitsAll the initial velocity vectors in the figure have the same magnitude, and the release point (with position vector \(\vec r_i\)) is the same for all the orbits, so they all have the same energy; inde...All the initial velocity vectors in the figure have the same magnitude, and the release point (with position vector \(\vec r_i\)) is the same for all the orbits, so they all have the same energy; indeed, you can check that the semimajor axis of the two ellipses is the same as the radius of the circle, as required by Equation (\ref{eq:10.14}).
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/02%3A_Moments_of_Inertia/2.20%3A_Ellipses_and_EllipsoidsThe distribution of mass around the minor axis is the same as for a circular lamina of radius \( a\), and therefore the moment \( B \) is the same as for the circular lamina, namely \( B = \frac{1}{4}...The distribution of mass around the minor axis is the same as for a circular lamina of radius \( a\), and therefore the moment \( B \) is the same as for the circular lamina, namely \( B = \frac{1}{4} ma^2 \). The moments of inertia of an elliptical ring of mass \(m\) and semi major and semi minor axes \(a\) and \(b\) are \(c_1ma^2 \) about the minor axis and \( c_2ma^2 \) about the major axis, where \(c_1 \) and \(c_2 \) are shown as functions of \(b/a \).
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Astronomy_2e_(OpenStax)/03%3A_Orbits_and_Gravity/3.02%3A_The_Laws_of_Planetary_MotionTycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of pl...Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period (P) and the semimajor axis (a) of an orbit is given by \(P^2 = a^3\) (when a is in units
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.02%3A_The_Ellipseellipse is the locus of a point that moves such that the sum of its distances from two fixed points called the foci is constant. An ellipse can be drawn by sticking two pins in a sheet of paper, tying...ellipse is the locus of a point that moves such that the sum of its distances from two fixed points called the foci is constant. An ellipse can be drawn by sticking two pins in a sheet of paper, tying a length of string to the pins, stretching the string taut with a pencil, and drawing the figure that results. During this process, the sum of the two distances from pencil to one pin and from pencil to the other pin remains constant and equal to the length of the string.
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/10%3A_Gravity/10.01%3A_The_Inverse-Square_LawSuppose that, at some time \(t_A\), the particle is at point A, and a time \(\Delta t\) later it has moved to A\(^{\prime}\) . The area “swept” by its position vector is shown in grey in the figure, a...Suppose that, at some time \(t_A\), the particle is at point A, and a time \(\Delta t\) later it has moved to A\(^{\prime}\) . The area “swept” by its position vector is shown in grey in the figure, and Kepler’s second law states that it must be the same, for the same time interval, at any point in the trajectory; so, for instance, if the particle starts out at B instead, then in the same time interval \(\Delta t\) it will move to a point B\(^{\prime}\) such that the area of the “curved triangl…
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/02%3A_Conic_Sections/2.06%3A_The_General_Conic_SectionThus if \(x\) is first replaced with \(x + \bar{g} / \bar{c}\) and \(y\) with \(y + \bar{f}/\bar{c}\), and then the new \(x\) is replaced with \(x \cos θ − y \sin θ\) and the new \(y\) with \(x \sin θ...Thus if \(x\) is first replaced with \(x + \bar{g} / \bar{c}\) and \(y\) with \(y + \bar{f}/\bar{c}\), and then the new \(x\) is replaced with \(x \cos θ − y \sin θ\) and the new \(y\) with \(x \sin θ + y \cos θ\), the Equation will take the familiar form of a conic section with its major or transverse axis coincident with the \(x\) axis and its centre at the origin.
