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Kepler's laws of planetary motion

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These improved the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits with epicycles with elliptical trajectories, and explaining how planetary velocities vary.The laws state that:

1.The orbit of a planet is an ellipse with the Sun at one of the two foci.

2.A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3.The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars.

From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.
The second law helps to establish that when a planet is closer to the Sun, it travels faster.
The third law expresses that the further a planet is from the Sun, the longer its orbit, and vice versa.
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.

History

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.Kepler's third law was published in 1619.
Kepler had believed in the Copernican model of the solar system, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury.
His first law reflected this discovery.

Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.
The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour.
As the century proceeded it became more widely accepted.
The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction.
Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

Kepler's laws Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

1.
2.

2.Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = rmin and for θ = 180°, r = rmax.

First law of Kepler (law of orbits)

The orbit of every planet is an ellipse with the Sun at one of the two foci.

Mathematically, an ellipse can be represented by the formula:

where {\displaystyle p}p is the semi-latus rectum, ε is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and θ is
the angle to the planet's current position from its closest approach, as seen from the Sun. So (r, θ) are polar coordinates.

For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

At θ = 0°, perihelion, the distance is minimum

At θ = 90° and at θ = 270° the distance is equal to {\displaystyle p}p.

At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

The semi-major axis a is the arithmetic mean between rmin and rmax:

The semi-minor axis b is the geometric mean between rmin and rmax

The semi-latus rectum p is the harmonic mean between rmin and rmax:

The eccentricity ε is the coefficient of variation between rmin and rmax

The area of the ellipse is

The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr2.

Second law of Kepler

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area.

In a small time {\displaystyle dt\,}dt\, the planet sweeps out a small triangle having base line {\displaystyle r\,}r\, and height {\displaystyle r\,d\theta }r\,d\theta and area {\displaystyle dA={\frac {1}{2}}\cdot r\cdot r\,d\theta }{\displaystyle dA={\frac {1}{2}}\cdot r\cdot r\,d\theta } and so the constant areal velocity is

The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity.

In a small time {\displaystyle dt\,}dt\, the planet sweeps out a small triangle having base line {\displaystyle r\,}r\, and height {\displaystyle r\,d\theta }r\,d\theta and area

and so the constant areal velocity is

The area enclosed by the elliptical orbit is \pi ab.\, So the period {\displaystyle P\,}P\, satisfies

and the mean motion of the planet around the Sun

n={2pi/ P}

satisfies

Third law of Kepler

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.
So it was known as the harmonic law.

Using Newton's Law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:

Then, expressing the angular velocity in terms of the orbital period and then rearranging, we find Kepler's Third Law:

A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass.
This results in replacing a circular radius,(r), with the elliptical semi-major axis,(a), as well as replacing the large mass(M) with (M+m)(M+m).
However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:

where(M) is the mass of the Sun,(m) is the mass of the planet, and (G) is the gravitational constant, (T) is the orbital period and (a) is the elliptical semi-major axis.

Data used by Kepler (1618)

Euler's laws of motion

In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion.
They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.

Euler's first law

Euler's first law states that the linear momentum of a body, p (also denoted G) is equal to the product of the mass of the body m and the velocity of its center of mass v cm

Internal forces between the particles that make up a body do not contribute to changing the total momentum of the body as there is an equal and opposite force resulting in no net effect. The law is also stated as:

where
is the acceleration of the centre of mass and
is the total applied force on the body.

This is just the time derivative of the previous equation (m is a constant).

Euler's second law

Euler's second law states that the rate of change of angular momentum L (sometimes denoted H) about a
point that is fixed in an inertial reference frame (often the mass center of the body),
is equal to the sum of the external moments of force (torques)
acting on that body M (also denoted τ or Γ) about that point:

Note that the above formula holds only if both M and L are computed with respect to a fixed inertial frame or a frame parallel to
the inertial frame but fixed on the center of mass.
For rigid bodies translating and rotating in only two dimensions, this can be expressed as:

where rcm is the position vector of the center of mass with respect to the point about which moments are summed, α is
the angular acceleration of the body about its center of mass, and I is the moment of inertia of the body about its center of mass.
See also Euler's equations (rigid body dynamics).

Explanation and Derivation

The distribution of internal forces in a deformable body are not necessarily equal throughout, i.e.
The stresses vary from one point to the next.
This variation of internal forces throughout the body is governed by Newton's second law of motion of conservation of linear momentum and angular momentum, which for their simplest use are applied to a mass particle but are extended in continuum mechanics to a body of continuously distributed mass.
For continuous bodies these laws are called Euler's laws of motion.
If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion, then Euler's equations can be derived from Newton's laws.
Euler's equations can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle distribution.

The total body force applied to a continuous body with mass m, mass density ρ, and volume V, is the volume integral integrated over the volume of the body:

where b is the force acting on the body per unit mass (dimensions of acceleration, misleadingly called the "body force"), and dm = ρ dV is an infinitesimal mass element of the body.

Body forces and contact forces acting on the body lead to corresponding moments (torques) of those forces relative to a given point. Thus, the total applied torque M about the origin is given by

where MB and MC respectively indicate the moments caused by the body and contact forces.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) acting on the body can be given as the sum of a volume and surface integral:

where t = t(n) is called the surface traction, integrated over the surface of the body, in turn n denotes a unit vector normal and directed outwards to the surface S.

Let the coordinate system (x1, x2, x3) be an inertial frame of reference, r be the position vector of a point particle in the continuous body with respect to the origin of the coordinate system, and
be the velocity vector of that point.

Euler's first axiom or law (law of balance of linear momentum or balance of forces) states that in an inertial frame the time rate of change of linear momentum p of an arbitrary portion of a continuous body is
equal to the total applied force F acting on that portion, and it is expressed as

Euler's second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum L of an arbitrary portion of a continuous body is equal to the total applied torque M acting on that portion, and it is expressed as

Where {v} is the velocity,V the volume, and the derivatives of p and L are material derivatives.