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- https://phys.libretexts.org/Bookshelves/University_Physics/Mechanics_and_Relativity_(Idema)/06%3A_General_Planar_Motion/6.04%3A_Kepler's_LawsThe fact that the planets move in elliptical orbits was first discovered by Kepler, based on observational data alone (he didn’t have the benefit, as we do, of living after Newton and thus knowing abo...The fact that the planets move in elliptical orbits was first discovered by Kepler, based on observational data alone (he didn’t have the benefit, as we do, of living after Newton and thus knowing about Newton’s law of gravity). Kepler summarized his observational facts in three laws, which we can, with the benefit of hindsight, prove to be corollaries of Newton’s laws.
- 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/Muhlenberg_College/MC%3A_Physics_121_-_General_Physics_I/13%3A_Gravitation/13.06%3A_Kepler's_Laws_of_Planetary_MotionJohannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws: ...Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws: Kepler’s first law states that every planet moves along an ellipse. Kepler’s second law states that a planet sweeps out equal areas in equal times. Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit.
- 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/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/13%3A_Gravitation/13.06%3A_Kepler's_Laws_of_Planetary_MotionJohannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws: ...Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws: Kepler’s first law states that every planet moves along an ellipse. Kepler’s second law states that a planet sweeps out equal areas in equal times. Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit.
- 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)/09%3A_The_Two_Body_Problem_in_Two_Dimensions/9.01%3A_Kepler's_LawsKepler’s law of planetary motion are as follows: 1. Every planet moves around the Sun in an orbit that is an ellipse with the Sun at a focus. 2. The radius vector from Sun to planet sweeps out equal ...Kepler’s law of planetary motion are as follows: 1. Every planet moves around the Sun in an orbit that is an ellipse with the Sun at a focus. 2. The radius vector from Sun to planet sweeps out equal areas in equal time. 3. The squares of the periods of the planets are proportional to the cubes of their semi major axes.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Celestial_Mechanics_(Tatum)/09%3A_The_Two_Body_Problem_in_Two_Dimensions/9.04%3A_Kepler's_First_and_Third_Laws_from_Newton's_Law_of_GravitationThe angular momentum per unit mass of \(m\) relative to the centre of mass is , \(\sqrt{G\mathfrak{M} l_2}\) where \(l_2\) is the semi latus rectum of the orbit of \(m\) relative to the centre of mass...The angular momentum per unit mass of \(m\) relative to the centre of mass is , \(\sqrt{G\mathfrak{M} l_2}\) where \(l_2\) is the semi latus rectum of the orbit of \(m\) relative to the centre of mass, and it is \(\sqrt{G\textbf{M} l}\) relative to \(M\), where \(l\) is the semi latus rectum of the orbit of \(m\) relative to \(M\).