Dwarf Planet

2009-04-13T17:05:07Z

Glenn: /* Table */ Haumea from http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=136108; Makemake inclination from Wikipedia

[[Image:Pluto sri.jpg|thumb|60 px|right|[[Pluto]]]]
A '''dwarf planet''' is a satellite of the [[Sun]] that meets the following criteria now set by the [[International Astronomical Union]]:{{Cquote|(a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared the neighbourhood around its orbit, and (d) is not a satellite.<ref>"[http://www.iau.org/iau0602.423.0.html IAU0602: the Final IAU Resolution on the Definition of 'Planet' Ready for Voting]," ''International Astronomical Union'', 2005. Accessed January 14, 2008.</ref>}}

The definition was adopted on 24 August, 2006, as part of a three-way classification of bodies orbiting the Sun.<ref>http://www.iau.org/static/resolutions/Resolution_GA26-5-6.pdf</ref> This classification defines [[planet|planets]] as "a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape and (c) has cleared the neighborhood around its orbit."

Next follows dwarf planets, followed by Small Solar System Bodies, which are defined as "celestial bodies that are not massive enough to be rounded by their own gravity."

== Eris ==
: ''Main Article: [[Eris]]''
[[Image:Eris and Dysnomia.jpg|thumb|200px|right|[[Eris]] and its satellite, [[Dysnomia]].]]

The concept of "dwarf planet" properly dates to the discovery of the scatter-disk object named [[Eris]] on January 5, 2005. With the confirmation of the identification of Eris, a formal debate began on the subject of what does, and what does not, constitute a [[planet]]. The community of [[Astronomy|astronomers]] decided that relying on history and tradition simply would not serve. The eight celestial bodies that now remain under the present definition of ''[[planet]]'' do not share their orbits with any other objects. The objects now called dwarf planets do not have this distinction. And yet these objects are significantly heavier than mere [[asteroid]]s, in that they are so heavy that their own weight, and more particularly their own gravity, forces them to assume the round or nearly-round shape that all planets have.

Eris has another distinction that forced the debate: it is more massive even than [[Pluto]] (by twenty-seven percent) and is therefore the heaviest dwarf planet yet found. The mass of Eris is inferred from the orbital parameters, including the [[Apsis|apsides]] and period, of the small body that orbits Eris. If Pluto could still be called a planet, then Eris deserved that distinction as well.

In 2006, the International Astronomical Union settled the issue. They determined that Eris was not a planet, and neither was Pluto. But the criteria they set also provoked yet another reassessment of the status of [[Ceres]], the largest and first-discovered object in the [[asteroid]] belt. Upon consideration, the IAU declared that Ceres was a dwarf planet as well.

Today the list of dwarf planets is limited to these five: Ceres, Pluto, Haumea, Makemake and Eris.

== Table ==
[[Image:Ceres.jpg|thumb|150px|right|Ceres]]List of all known dwarf planets, from the innermost to the outermost:
{| class="wikitable"
|-
! Name
! [[Apsis|Perihelion]]
! [[Apsis|Aphelion]]
! [[Orbital eccentricity|Eccentricity]]
! [[Orbital period|Sidereal year]]
! [[Inclination]]
! [[Mass]]
! [[Sidereal day]]
|-
| [[Ceres]]
| 2.545 AU
| 2.987 AU
| 0.0798
| 4.599 a
| 10.587 °
| 0.0158% earth
| 0.378 da
|-
| [[Pluto]]
| 29.658 AU
| 49.305 AU
| 0.249
| 248.09 a
| 17.142 °
| 0.218% earth
| -6.387 da
|-
| [[Haumea]]
| 34.623 AU
| 51.538 AU
| 0.196
| 282.77 a
| 28.22 °
| (4.2 ± 0.1)×10<sup>21</sup> kg
| Unknown
|-
| [[Makemake]]
| 38.509 AU
| 53.074 AU
| 0.159
| 309.88 a
| 28.96 °
| ~4 × 10<sup>21</sup> kg
| Unknown
|-
| [[Eris]]
| 37.77 AU
| 97.56 AU
| 0.442
| 577 a
| 44.187 °
| 0.278% earth
| 0.333 da
|}

==Plutoids==

The name '''plutoid''' was proposed by the members of the IAU Committee on Small Body Nomenclature (CSBN), to describe celestial bodies "in orbit around the Sun at a semi-major axis greater than that of Neptune that have sufficient mass for their self-gravity to overcome rigid body forces so that they assume a hydrostatic equilibrium (near-spherical) shape, and that have not cleared the neighborhood around their orbit." <ref>http://solarsystem.nasa.gov/planets/profile.cfm?Object=Dwarf&Display=OverviewLong</ref>

The four known and named plutoids are Pluto, Haumea, Makemake and Eris.

It is important to note that while all plutoids are dwarf planets, not all dwarf planets are plutiods. Ceres, for example, is a dwarf planet, but not a plutoid, as its orbit lies well within that of Neptune.

== References ==
<references/>
{{Solarsystem}}

== See Also ==
* [[Planet]]
* [[Eris]]
* [[Pluto]]
* [[Ceres]]
* [[Haumea]]
* [[Makemake]]

[[Category:Dwarf planets]]

Cyclotomic polynomials

2009-04-13T16:17:50Z

Glenn: Order from lowest to highest (1,2,3,6 instead of 6,3,2,1)

'''Cyclotomic''' (or "circle dividing") '''polynomials''' are the irreducible factors of <math>z^n - 1\!</math> when the polynomial coefficients are restricted to the field of [[rational number]]s. For example

:<math> z^6-1 = (z-1) (z+1) (z^2+z+1) (z^2-z+1) </math>.

The cyclotomic polynomials are often written as <math>\Phi_n(z)</math>. The polynomial <math>z^n - 1\!</math> has exactly as many of these factors as there are integer divisors of n:

:<math> z^6 - 1 = \Phi_1(z)\Phi_2(z)\Phi_3(z)\Phi_6(z) </math>.

==Finding cyclotomic polynomials==

===Recursive method===

Cyclotomic polynomials are defined by the equation

:<math>z^n - 1 = \prod_{d\,\mid\,n} \Phi_d(z)</math>.

Solving for <math>\Phi_n(z) </math>, one obtains

:<math>\Phi_n(z) = \frac {z^n - 1} {\prod_{(d \ne n)\,\mid\,n} \Phi_d(z)} </math>.

One can then construct any polynomial recursively, given that <math> \Phi_1(z) = z - 1</math>:

:<math>\Phi_2(z) = \frac {z^2 - 1} {z-1} = z + 1 </math>

:<math>\Phi_3(z) = \frac {z^3 - 1} {z-1} = z^2+z+1 </math>

:<math>\Phi_6(z) = \frac {z^6 - 1} {(z-1) (z+1) (z^2 + z + 1)} = z^2-z+1 </math>.

===Direct method===

For larger values of n, it is easier to use the direct formula, based on the Möbius inversion of the above formula:

:<math>\Phi_n(z)=\prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)},</math>

where μ is the [[Möbius function]]. For example, one would follow these steps for n = 60:

:'''1.''' Factor n.

::<math> 60 = 2^2*3*5 </math>

:'''2.''' Determine the square-free divisors of n, and classify them according to whether their number of prime factors are even or odd.

::<math> 30 = 2*3*5 </math> (Odd)
::<math> 15 = 3*5 </math> (Even)
::<math> 10 = 2*5 </math> (Even)
::<math> 6 = 2*3 </math> (Even)
::<math> 5= 5 </math> (Odd)
::<math> 3= 3 </math> (Odd)
::<math> 2= 2 </math> (Odd)
::<math> 1= 1 </math> (Even, since 1 is not a prime)

:'''3.''' Form the factors <math> z^{n/d} - 1 </math>, putting the factors from the even class in the numerator and those from the odd class in the denominator.

::<math> \Phi_{60}(z) = \frac { (z^{60}-1) (z^{10}-1) (z^6-1) (z^4-1) } { (z^{30}-1) (z^{20}-1) (z^{12}-1) (z^2-1) } </math>

:::<math> = \frac { (z^{30}+1) (z^2+1) } { (z^{10}+1) (z^6+1) } </math>

:::<math> = \frac { z^{20}-z^{10}+1 } { z^4-z^2+1 } </math>

:::<math> = z^{16}+z^{14}-z^{10}-z^8-z^6+z^2+1 </math>.

===Construction from complex roots===

It is also possible to construct a cyclotomic polynomial from the complex roots of 1:

:<math>\Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-e^{\frac {2\pi k} {n} j})\;</math>

where <math>\phi(n)</math> is [[Euler's totient function]]. For example,

:<math>\Phi_6(z) = (z-e^{\frac {\pi} {3} j})(z-e^{\frac {5\pi} {3} j}) = z^2-z+1.</math>

==Some special cases==
When p and q are odd primes,

:<math>\Phi_p(z)=\frac{z^p-1}{z-1}=\sum_{k=0}^{p-1} z^k</math>

:<math>\Phi_{2p}(z)=\frac{z^p+1}{z+1}=\sum_{k=0}^{p-1} (-z)^k</math>

:<math>\Phi_{pq}(z)=\frac{(z^{pq}-1) (z-1)}{(z^p-1) (z^q-1)}=\frac {\sum_{k=0}^{q-1} z^{pk} } {\sum_{k=0}^{q-1} z^k}</math>

==Galois Theory==

The [[Galois group]] associated to the nth cyclotomic polynomial is <math>(\mathbb{Z}/n\mathbb{Z})^*</math>, the group of cyclic units modulo n. Cyclotomic fields play an important part in [[algebra]]ic [[number theory]] and can be used to show that certain constructions posited by ancient Greek geometers are impossible, such as trisecting the angle and constructing certain regular polygons.

[[category:algebra]]

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Dwarf Planet

Cyclotomic polynomials