The polar coordinate system is an alternative to the cartesian coordinate system. Whereas points in the cartesian system are denoted (x,y), points in the polar coordinate system are denoted (r,theta). This coordinate system defines points not on a grid, but rather more on a continuous set of circles. A point at (r,theta) would be located on the circle with radius r at the position displaced theta (degrees or radians, usually radians) from the right horizontal axis.

The right horizontal axis represents 0 degrees, and you would draw the circle counterclockwise from 0 degrees to find the position of the angle. Likewise, if the angle were negative, you would draw it clockwise. Each quadrant is a total of 90 degrees or pi/2 radians. When you reach 360 degrees or 2pi radians, you are back at the right horizontal axis. In this way, if theta is greater than 360 (or less than 0 for that matter), it can be converted to another number that fits in that range, which makes for easier graphing.

This coordinate system has only been useful, at least for me, in Calculus. Some integrals are only solvable if you use a change of variables, which sometimes involves exchanging x and y for r and theta. The equations for such are below:

r=(x2+y2)1/2 <-- Comes from our good friend Pythagorus
theta=tan-1(y/x)

x=r cos theta
y=r sin theta

Polar is also a term used to define atomic charges.
When atoms combine in certain ways, electrons are shared unequally. Since electrons have a negative charge, one side of the molecule becomes negatively charged, making the other side positively charged.
Examples of polar molecules include:

Polars (pronounced "pole-ars", like pulsars) are white dwarf stars in accreting binary star systems that emit a large amount of polarised light, where the polarization is caused by their strong magnetic fields. The magnetic fields are also responsible for their dynamical behavior; they have such strong magnetic fields (from 10 to 230 megagauss) that the rotation rate of the white dwarf is tidally locked to the orbital period of the binary. The true polars, also known as AM Herculis stars, accrete matter from the secondary star overfilling its Roche lobe. But instead of forming an accretion disk, matter flows from the secondary along the magnetic field lines of the white dwarf, directly onto the magnetic poles of the white dwarf. From there, the matter spreads out over the surface of the star, increasing the mass of the white dwarf over time.

As a class, the polars all have very short orbital periods -- most below two hours, and all below four hours. In these systems, the "magnetic radius" rμ, where the magnetic force equals the ram pressure of infalling gas, is a significant fraction of the binary separation. So what happens is that gas flowing off of the secondary star flows into a circular orbit around the white dwarf, but at a few hundred thousand kilometers from the white dwarf, it gets channeled into these magnetic streams.

When matter falls onto the white dwarf, it really falls hard. White dwarfs have the entire mass of a star packed into a sphere a few thousand miles across, so the force of gravity at the surface is a thousand times that on the surface of the Sun, and hundreds of thousands of times stronger than the force of gravity on Earth. So when matter falls along the magnetic field lines, it is moving very quickly. The matter falls in blobs, rather than a continuous flow; the little blobs fall apart and slow down in the upper atmosphere of the white dwarf where they emit lots of X-rays and optical light. The bigger blobs actually make it to the surface where they impact moving at about 1 percent of the speed of light. The gas releases all of this kinetic energy as heat, and emits blackbody radiation at a temperature of several hundred million kelvins. So the spectra of the AM Herculis stars are chock full of interesting features that change drastically over just a few minutes. With all this activity, the polars are quite variable stars, and their brightnesses can change by a few magnitudes over the course of the orbital period.

Although the polars don't necessarily have outbursts like the dwarf, recurrent, or classical novae, they may eventually undergo a nova explosion on the surface of the star, or explode as type I supernova if they accrete enough mass to surpass the Chandrasekhar limit. No one really knows if any of the known polars are close to doing this, however, so don't hold your breath -- it could take millions of years.

The polars have poor cousins called intermediate polars, or DQ Herculis stars, which (probably) have weaker magnetic fields (5 megagauss or less). These stars also funnel matter down onto the white dwarf via the magnetic field lines, but the infalling matter has enough angular momentum to form a proper accretion disk within the system. The magnetic field then sucks matter from the inner edge of the accretion disk down to the star. So in the intermediate polars, you can have a bright hotspot where matter falls from the companion star onto the disk, but you don't have a boundary layer in the inner disk to generate heat and light. The intermediate polars are asynchronous rotators -- the white dwarf rotates faster than the binary star orbit so they aren't tidally locked. Again, this is probably because the magnetic field is weaker, so the interaction of the white dwarf's field with that of the secondary star doesn't produce strong torque.

There are a few dozen known polars in the Milky Way. Many of these were found by X-ray observatories like the ROSAT satellite, mainly because of the strong X-ray emission they can give out. However, many were known from optical observations simply because they were so prominently variable, and the strong magnetic fields were easily detected with polarimetry. AM Herculis, the class prototype, can be observed with a moderate-sized backyard telescope (8-12 inches/20-30 cm). It is a magnitude 12.5 object, located in the constellation Hercules (α 18h 16m 13.4s, δ +49° 52' 3.1').

Po"lar (?), a. [Cf. F. polaire. See Pole of the earth.]

1.

Of or pertaining to one of the poles of the earth, or of a sphere; situated near, or proceeding from, one of the poles; as, polar regions; polar seas; polar winds.

2.

Of or pertaining to the magnetic pole, or to the point to which the magnetic needle is directed.

3. Geom.

Pertaining to, reckoned from, or having a common radiating point; as, polar coordinates.

Polar axis, that axis of an astronomical instrument, as an equatorial, which is parallel to the earths axis. -- Polar bear Zool., a large bear (Ursus, ∨ Thalarctos, maritimus) inhabiting the arctic regions. It sometimes measures nearly nine feet in length and weighs 1,600 pounds. It is partially amphibious, very powerful, and the most carnivorous of all the bears. The fur is white, tinged with yellow. Called also White bear. See Bear. -- Polar body, cell, ∨ globule Biol., a minute cell which separates by karyokinesis from the ovum during its maturation. In the maturation of ordinary ova two polar bodies are formed, but in parthogenetic ova only one. The first polar body formed is usually larger than the second one, and often divides into two after its separation from the ovum. Each of the polar bodies removes maternal chromatin from the ovum to make room for the chromatin of the fertilizing spermatozoon; but their functions are not fully understood. -- Polar circles Astron. & Geog., two circles, each at a distance from a pole of the earth equal to the obliquity of the ecliptic, or about 23° 28�xb7;, the northern called the arctic circle, and the southern the antarctic circle. -- Polar clock, a tube, containing a polarizing apparatus, turning on an axis parallel to that of the earth, and indicating the hour of the day on an hour circle, by being turned toward the plane of maximum polarization of the light of the sky, which is always 90° from the sun. -- Polar coordinates. See under 3d Coordinate. -- Polar dial, a dial whose plane is parallel to a great circle passing through the poles of the earth. Math. Dict. -- Polar distance, the angular distance of any point on a sphere from one of its poles, particularly of a heavenly body from the north pole of the heavens. -- Polar equation of a linesurface, an equation which expresses the relation between the polar coordinates of every point of the line or surface. -- Polar forces Physics, forces that are developed and act in pairs, with opposite tendencies or properties in the two elements, as magnetism, electricity, etc. -- Polar hare Zool., a large hare of Arctic America (Lepus arcticus), which turns pure white in winter. It is probably a variety of the common European hare (L. timidus). -- Polar lights, the aurora borealis or australis. -- Polar, ∨ Polaric, oppositioncontrast Logic, an opposition or contrast made by the existence of two opposite conceptions which are the extremes in a species, as white and black in colors; hence, as great an opposition or contrast as possible. -- Polar projection. See under Projection. -- Polar spherical triangle Spherics, a spherical triangle whose three angular points are poles of the sides of a given triangle. See 4th Pole, 2. -- Polar whale Zool., the right whale, or bowhead. See Whale.

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© Webster 1913.


Po"lar (?), n. Conic Sections

The right line drawn through the two points of contact of the two tangents drawn from a given point to a given conic section. The given point is called the pole of the line. If the given point lies within the curve so that the two tangents become imaginary, there is still a real polar line which does not meet the curve, but which possesses other properties of the polar. Thus the focus and directrix are pole and polar. There are also poles and polar curves to curves of higher degree than the second, and poles and polar planes to surfaces of the second degree.

 

© Webster 1913.

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