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- 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 P2=a3 (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 P2=a3
- 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<<RE.
- 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 ME and RE 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 →ri) 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 →ri) 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 (???).
- 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 P2=a3 (when a is in units
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/25%3A_Celestial_Mechanics/25.04%3A_Energy_Diagram_Effective_Potential_Energy_and_OrbitsThe effective potential energy describes the potential energy for a reduced body moving in one dimension. (Note that the effective potential energy is only a function of the variable r and is independ...The effective potential energy describes the potential energy for a reduced body moving in one dimension. (Note that the effective potential energy is only a function of the variable r and is independent of the variable θ ). The constant r0 is independent of the energy and from Equation (25.3.14) as the energy of the single body increases, the eccentricity increases, and hence from Equation (25.4.26), the distance of closest approach gets smaller (Figure 25.5).
- 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 tA, the particle is at point A, and a time Δt later it has moved to A′ . The area “swept” by its position vector is shown in grey in the figure, a...Suppose that, at some time tA, the particle is at point A, and a time Δt later it has moved to A′ . 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 Δt it will move to a point B′ such that the area of the “curved triangl…
- 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/13.02%3A_Kepler's_LawsThe 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 ins...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 Δt it will move to a point B′ such that the area of the “curved triangle” OBB′ equals the area of OAA′.
