Johannes Kepler's primary contributions to astronomy/astrophysics were the three laws of planetary motion. Kepler derived these laws, in part, by studying the observations of Brahe. Isaac Newton would later design his laws of motion and universal gravitation and verify that Kepler's laws could be derived from them. The generic term for an orbiting object is "satellite". Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ... Astronomy (Greek: αστρονομία = άστρον + νόμος, literally, law of the stars) is the science involving the observation and explanation of events occurring beyond the Earth and its atmosphere. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties ( luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Observation basically means watching something and taking note of anything it does. ... Tycho Brahe (December 14, 1546 Knudstrup, Denmark – October 24, 1601 Prague, Bohemia (now Czech Republic)) was a Danish nobleman, well known as an astronomer/astrologer (the two were not yet distinct) and alchemist. ... Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... -1... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ... For other uses, please see Satellite (disambiguation) A satellite is an object that orbits another object (known as its primary). ...

Kepler's laws of planetary motion

• Kepler's first law (1609): The orbit of a planet about a star is an ellipse with the star at one focus.

• Kepler's second law (1609): A line joining a planet and its star sweeps out equal areas during equal intervals of time.

• Kepler's third law (1618): The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis.

For other meanings of the term orbit, see orbit (disambiguation) In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... For alternate meanings see star (disambiguation) Hundreds of stars are visible in this image taken by the Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. ... In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. ... The word focus (pl. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... This article explains the meaning of area as a physical quantity. ... (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ... This article is about proportionality, the mathematical relation. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. ...

Kepler's first law

The orbit of a planet about a star is an ellipse with the star at one focus. For other meanings of the term orbit, see orbit (disambiguation) In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... For alternate meanings see star (disambiguation) Hundreds of stars are visible in this image taken by the Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. ... The word focus (pl. ...

There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near apocentre than near pericentre. This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

Connection with Newton's laws

Newton proposed that "every object in the universe attracts every other object along a line of the centers of the objects proportional to each objects mass, and inversely proportional to the square of the distance between the objects."

This section proves that Kepler's first law is consistent with Newton's laws of motion. We begin with Newton's law F=ma:-1...

Here we express F as the product of its magnitude and its direction. Recall that in polar coordinates: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

In component form we have:

Now consider the angular momentum: In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ...

So:

where is the angular momentum per unit mass. Now we substitute. Let: In astrodynamics specific relative angular momentum () of orbiting body () relative to central body () is the relative angular momentum of per unit mass. ...

The equation of motion in the direction becomes:

Newton's law of gravitation states that the central force is inversely proportional to the square of the distance so we have:

where k is our proportionality constant.

This differential equation has the general solution:

Replacing u with r and letting θ₀=0:

.

This is indeed the equation of a conic section with the origin at one focus. Q.E.D. In mathematics, a conic section (or just conic) is a curved locus of points, fby intersecting a cone with a plane. ...

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time. The word line apparently derives from the Latin linum, meaning flax plant from which linen is produced; at one time, a stretched linen thread was the most reliable way to determine a straight line. ... This article explains the meaning of area as a physical quantity. ... (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. ...

This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day. A day is any of several different units of time. ... The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution... (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... This article explains the meaning of area as a physical quantity. ...

As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. Planets move fastest when at perihelion and slowest when at aphelion. For distance between people, see proxemics. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... This article is about velocity in physics. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

This law was developed, in part, from the observations of Brahe that indicated that the velocity of planets was not constant. Tycho Brahe (December 14, 1546 Knudstrup, Denmark – October 24, 1601 Prague, Bohemia (now Czech Republic)) was a Danish nobleman, well known as an astronomer/astrologer (the two were not yet distinct) and alchemist. ...

This law corresponds to the angular momentum conservation law in the given situation. In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

Proof of Kepler's second law:

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum of a point mass with mass m and velocity is : In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ...

.

where is the position vector of the particle.

Since , we have:

taking the time derivative of both sides:

since the cross product of parallel vectors is 0. We can now say that is constant. In mathematics, the cross product is a binary operation on vectors in three dimensions. ... A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this...

The area swept out by the line joining the planet and the sun, is half the area of the parallelogram formed by and . A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... A Sun is the star at the centre of a solar system. ... This article explains the meaning of area as a physical quantity. ... A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides. ...

Since is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law (harmonic law)

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis. In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ... The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. ...

P² ~ a³

P = object's sidereal period in years

a = object's semimajor axis, in AU

Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional. The astronomical unit (AU or au or a. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

See the actual figures: attributes of major planets. Mosaic of the planets of the solar system, excluding Pluto, and including Earths Moon. ...

Newton would modify this third law, noting that the period is also affected by the orbiting body's mass, however typically the central body is so much more massive that the orbiting body's mass may be ignored. (See below.) Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ...

Applicability

The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a satellite revolving around Earth. For other uses, please see Satellite (disambiguation) A satellite is an object that orbits another object (known as its primary). ...

Application

Assume an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points: (This page refers to eccitricity in astrodynamics. ...

• c center of auxiliary circle and ellipse
• s sun (at one focus of ellipse);
• p the planet
• z perihelion
• x is the projection of the planet to the auxiliary circle; then
• y is a point on the circle such that

and three angles measured from perihelion:

• true anomaly , the planet as seen from the sun
• eccentric anomaly , x as seen from the centre
• mean anomaly , y as seen from the centre

In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ... The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. ...

Then

area cxz = area cxs + area sxz = area cxs + area cyz

giving Kepler's equation

.

To connect E and T, assume r = length sp then

and rsinT = bsinE

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities: In mathematics, trigonometric identities (or trig identities for short) are equations involving trigonometric functions that are true for all values of the occurring variables. ...

(So far only laws of geometry have been used.)

Note that area spz is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined and so M is also proportional to time since perihelion—this is why it was introduced.

We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M. [1] (http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html) This page is about Lagrange reversion. ... Atmospheric characteristics Atmospheric pressure 0. ...

Kepler's understanding of the laws

Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. Newton, understanding that his third law of motion was related to Kepler's third law of planetary motion, devised the following: Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... -1...

where:

• P = object's sidereal period
• a = object's semimajor axis
• G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant
• m₁ = mass of object 1
• m₂ = mass of object 2
• π = mathematical constant pi

Astronomers doing celestial mechanics often use units of years, AU, G=1, and solar masses, and with m₂<<m₁, this reduces to Kepler's form. SI units may also be used directly in this formula. The orbital period is the time it takes a planet (or another object) to make one full orbit. ... In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... The minuscule, or lower-case, pi The mathematical constant π is commonly used in mathematics and physics. ... SI (disambiguation). ...

See also

• Circular motion
• Gravity
• Two-body problem
• Free-fall time