View source for Mathematics/Astronomy

[[Image:Superclusters atlasoftheuniverse.gif|thumb|250px|right|Our universe within 1 billion light-years (307 Mpc) of Earth is shown to contain the local [[w:supercluster|supercluster]]s, [[w:galaxy filament|galaxy filament]]s and voids. Credit: Richard Powell.]]
Although most of the mathematics needed to understand the information acquired through astronomical observation comes from physics, there are special needs from situations that intertwine mathematics with phenomena that may not yet have sufficient physics to explain the observations. These two uses of mathematics make '''mathematical astronomy''' a continuing challenge.

Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera, specifically its charge-coupled device (CCD) detector, that is attached to the telescope basically converts or counts photons or electrons and records a series of numbers (the counts) - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.

An initial use of mathematics in astronomy is counting [[Radiation astronomy/Entities|entities]], sources, or objects in the sky.

Objects may be counted during the daytime or night.

One use of mathematics is the calculation of distance to an object in the sky.

{{clear}}

==Notations==

'''Notation''': let the symbol <math>\oplus</math> indicate the [[Earth]].

'''Notation''': let the symbol '''ʘ''' or <math>\odot</math> indicate the [[Stars/Sun|Sun]].

'''Notation''': let the symbol <math> I_{\odot} </math> indicate the total solar irradiance.

'''Notation''': let the symbol <math> L_V </math> indicate the solar visible luminosity.

'''Notation''': let the symbol <math> L_{\odot} </math> indicate the solar bolometric luminosity.

'''Notation''': let the symbol <math> L_{bol} </math> indicate the solar bolometric luminosity.

'''Notation''': let the symbol <math>M_{bol}</math> represent the '''bolometric magnitude''', the total energy output.

'''Notation''': let the symbol <math>M_V</math> represent the '''visual magnitude'''.

'''Notation''': let the symbol <math> M_{\odot} </math> indicate the solar mass.

'''Notation''': let the symbol <math> Q_{\odot} </math> represent the net solar charge.

'''Notation''': let the symbol <math>R_\oplus</math> indicate the Earth's radius.

'''Notation''': let the symbol <math>R_J</math> indicate the radius of Jupiter.

'''Notation''': let the symbol <math>R_{\odot}</math> indicate the solar radius.

:{| border="1" cellpadding="5" cellspacing="0" align="none"
|-
! colspan="3" |'''Notational locations'''
|-
|'''Weight'''
|'''Oversymbol'''
|'''Exponent'''
|-
|'''Coefficient'''
|'''Variable'''
|'''Operation'''
|-
|'''Number'''
|'''Range'''
|'''Index'''
|-
|}

For each of the notational locations around the central '''Variable''', conventions are often set by consensus as to use. For example, '''Exponent''' is often used as an exponent to a number or variable: 2<sup>-2</sup> or x<sup>2</sup>.

In the '''Notation'''s at the top of this section, '''Index''' is replaced by symbols for the Sun (ʘ), [[Earth]] (<math>R_\oplus</math>), or can be for Jupiter (J) such as <math>R_J</math>.

A common '''Oversymbol''' is one for the average <math>\overline{Variable}</math>.

'''Operation''' may be replaced by a function, for example.

All notational locations could look something like

:{| border="0" cellpadding="3" cellspacing="0" align="none"
|-
|bx
|<math>-</math>
|x = n
|-
|a
|<math>\sum</math>
|f(x)
|-
|n
|→
|∞
|-
|}
where the center line means "a x Σ f(x)" for all added up values of f(x) when x = n from say 0 to infinity with each term in the sum before summation multiplied by bn, then divided by n for an average whenever n is finite.

==Abstractions==
{{main|Abstractions}}
A '''nomy''' (Latin ''nomia'') is a "system of [[Laws|law]]s governing or [the] sum of [[knowledge]] regarding a (specified) field."<ref name=Gove>{{ cite book
|author=
|title=Webster's Seventh New Collegiate Dictionary
|publisher=G. & C. Merriam Company
|location=Springfield, Massachusetts
|date=1963
|editor=Philip B. Gove
|pages=1221
|bibcode=
|doi=
|pmid=
|isbn=
}}</ref> ''Nomology'' is the "[[What is science?|science]] of physical and logical laws."<ref name=Gove/>

'''Def.''' the quality of dealing with ideas rather than events is called '''abstraction'''.

'''Def.''' the act of the theoretical way of looking at things; something that exists only in idealized form is called '''abstraction'''.

==Relations==
{{main|Relations}}
'''Notation''': let the relation symbol '''≠''' indicate that two expressions are different.

For example, 2 x 3 ≠ 5 x 7.

'''Notation''': let the relation symbol '''~''' represent '''similar to'''.

For example, depending on the scale involved, 7 ~ 8 on a scale of 10, 7/10 = 0.7 and 8/10 = 0.8. relative to numbers between 0.5 and 1.0, 0.7 ~ 0.8, but 0.2 ≁ 0.7.

Similarity may be close such as 0.7 ≈ 0.8, but 0.5 ~ 0.8. Or similarity may include equality, 5 ± 3 ≃ 4 ± 2. When the degree of equality is greater than the degree of similarity, the symbol ≅ is used. The reverse is represented by ≊.

==Differences==
{{main|Abstractions/Differences|Differences}}
Here's a [[Definitions/Theory#Theoretical definition|theoretical definition]]:

'''Def.''' an abstract relation between identity and sameness is called a '''difference'''.

'''Notation''': let the symbol <math>\Delta</math> represent '''difference in'''.

==Order==
{{main|Order}}
Ordering numbers may mean listing them in increasing value. For example, 2 is less than 3 so that in increasing order 2,3 is the list.

'''Notation''': let the symbol '''>''' represent '''greater than'''.

For example, the integer five (5) is greater than the integer (2): 5 > 2.

'''Notation''': let the symbol '''<''' represent '''less than'''.

For, example, 2 < 3.

==Numbers==
{{main|Numbers}}
:{|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20"
|<math>1, 2, 3,\ldots\!</math> || <math>\ldots,-2, -1, 0, 1, 2\,\ldots\!</math> || <math> -2, \frac{2}{3}, 1.21\,\!</math> || <math>-e, \sqrt{2}, 3, \pi\,\!</math> || <math>2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!</math>
|-
|[[Natural number]]s|| [[Integer]]s || [[w:Rational number|Rational number]]s || [[Real Numbers|Real number]]s || [[Complex Numbers|Complex number]]s
|}
[[Image:Avogadro's number in e notation.jpg|thumb|upright|200px|A calculator display showing an approximation to the [[w:Avogadro constant|Avogadro constant]] in E notation. Credit: [[commons:User:PRHaney|PRHaney]].]]
'''Scientific notation''' (more commonly known as '''standard form''') is a way of writing numbers that are too big or too small to be conveniently written in decimal form.  Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers.

{| class="wikitable" style="float:left"
|-
!Standard decimal notation
!Normalized scientific notation
|-
| 2
|{{val|2|e=0}}
|-
| 300
|{{val|3|e=2}}
|-
| 4,321.768
|{{val|4.321768|e=3}}
|-
| -53,000
|{{val|-5.3|e=4}}
|-
| 6,720,000,000
|{{val|6.72|e=9}}
|-
| 0.2
|{{val|2|e=-1}}
|-
| 0.000&nbsp;000&nbsp;007&nbsp;51
|{{val|7.51|e=-9}}
|}

A '''metric prefix''' or '''SI prefix''' is a [[w:unit prefix|unit prefix]] that precedes a basic unit of measure to indicate a [[w:decimal|decadic]] [[w:multiple (mathematics)|multiple]] or [[w:fraction (mathematics)|fraction]] of the unit. Each prefix has a unique symbol that is prepended to the unit symbol.

{{SI prefixes}}

A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes [[w:Significant figures#Identifying significant digits|indicated to be significant.]]

Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures—1, 2, 3, 0, and 4; the two zeroes serve only as placeholders and add no precision to the original number.

When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but all of the place holding zeroes are incorporated into the exponent. Following these rules, 1,230,400 becomes 1.2304 x 10<sup>6</sup>.

It is customary in scientific measurements to record all the significant digits from the measurements, for example, 1,230,400, but the measurement may have introduced an error which when calculated indicates the last significant digit has a range of values where the most likely one is the "4". The range may be 3-5 so that the last significant digit plus this error may be written as (4,1) meaning 4-1=3 and 4+1=5.

Another example of significant digits is the speed of all [[w:massless particle|massless particle]]s and associated [[w:field (physics)|field]]s—including [[w:electromagnetic radiation|electromagnetic radiation]] such as [[w:light|light]]—in vacuum ... [The most accurate value is] 299792.4562±0.0011 [km/s].<ref name="NIST heterodyne">{{cite journal
 |last1=Evenson 
 |first1=KM 
 |display-authors=etal
 |year=1972
 |title=Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser
 |journal=Physical Review Letters
 |volume=29
 |issue=19 |pages=1346–49
 |doi=10.1103/PhysRevLett.29.1346
|bibcode=1972PhRvL..29.1346E
}}</ref> The magnitude of the speed is 299792.4562 and the actual measured variation is ±0.0011 so that the last two significant digits "62" are most likely within a variation from "51" to "73".

Most [[w:calculator|calculator]]s and many [[w:computer program|computer program]]s present very large and very small results in E notation. The [[w:E|letter ''E'' or ''e'']] is often used to represent ''times ten raised to the power of'' (which would be written as "x&nbsp;10<sup>''b''</sup>") [where ''b'' represents a number] and is followed by the value of the exponent.

'''Def.''' any real number that cannot be expressed as a ratio of two integers is called an '''irrational number'''.

'''Def.''' incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof is called '''uncountable'''.

An uncountable set of numbers such as the irrational numbers lies somewhere between a finite set of numbers, for example, the set of natural factors of 6: {1,2,3,6}, and an infinite set of numbers such as the [[w:Natural number|natural number]]s.

'''Def.''' the branch of pure mathematics concerned with the properties of integers is called '''number theory'''.

==Arithmetics==
{{main|Arithmetics}}
Arithmetic involves the manipulation of numbers.

===Theory of arithmetic===

'''Def.''' the [[wikt:mathematics|mathematics]] of [[wikt:number|number]]s ([[wikt:integer|integer]]s, [[wikt:rational number|rational number]]s, [[wikt:real number|real number]]s, or [[wikt:complex number|complex number]]s) under the operations of [[wikt:addition|addition]], [[wikt:subtraction|subtraction]], [[wikt:multiplication|multiplication]], and [[wikt:division|division]] is called an '''arithmetic'''.

'''Def.''' a symbol ( = ) used in mathematics to indicate that two values are the same is called '''equals''', or '''equal to'''.

Consider the integers: 1 and 2. The statement, "1 + 2 = 3", contains the operation + (addition) and the relation = (equals).

===Exponentials===
{{main|Exponentials}}
The exponential can require a different operator of arithmetic.

The number '''<math>e</math>''' is an important [[w:mathematical constant|mathematical constant]], approximately equal to 2.71828, that is the base of the [[w:natural logarithm|natural logarithm]].<ref>Oxford English Dictionary, 2nd ed.: [http://oxforddictionaries.com/definition/english/natural%2Blogarithm natural logarithm]</ref>

:<math>e = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \frac{1}{1\cdot 2\cdot 3\cdot 4}+\cdots</math> 

: e<sup>2</sup> + e<sup>3</sup> ≠ e<sup>5</sup>. Yet
: e<sup>2</sup> x e<sup>3</sup> = e<sup>(2 + 3)</sup> = e<sup>5</sup>.

===Logarithms===
{{main|Logarithms}}
'''Def.''' For a number [...], the [exponent or power] to which a given ''base'' number must be raised in order to obtain [the number] is called a '''logarithm'''.

Consider 10<sup>3</sup>, the ''base'' number is 10. The exponent is 3. The number itself is 1000. Using logarithm notation
: <math>\log_{10} 1000 = 3.</math>

For 2<sup>4</sup>,
: <math> \log_{2} 16 = 4.</math>

For e<sup>4</sup>,
: <math> \ln_{e} e^4 = 4.</math>

'''Notation''': let the symbol '''dex''' represent the difference between powers of ten.

An order or factor of ten, dex is used both to refer to the function <math>dex(x) = 10^x</math> and the number of (possibly fractional) orders of magnitude separating two numbers.  When dealing with [[wikt:log|log]] to the base 10 transform of a number set, the transform of 10, 100, and 1 000 000 is <math>\log_{10}(10) = 1</math>, <math>\log_{10}(100) = 2</math>, and <math>\log_{10}(1 000 000) = 6</math>, so the difference between 10 and 100 in base 10 is 1 dex and the difference between 1 and 1 000 000 is 6 dex.

==Mathematics==
{{main|Mathematics}}
'''Def.''' an abstract representational system used in the study of [[wikt:number|number]]s, [[wikt:shape|shape]]s, [[wikt:structure|structure]] and [[wikt:change|change]] and the relationships between these concepts is called '''mathematics'''.

'''Def.''' the branches of mathematics used in the study of astronomy, [[astrophysics]] and cosmology is called '''astromathematics'''.

==Changes==
{{main|Changes}}
'''Def.''' the act or instance of making, or the process of becoming different is called '''change'''.

==Revolutions==
{{main|Changes/Revolutions|Revolutions}}
'''Def.''' the traversal of one body through an orbit around another body is called a '''revolution'''.

==Operations==
{{main|Operations}}
However, attempting to add 1 dome to 1 telescope may have little or no meaning. The operation of '''addition''' would be similar to the operation of '''construction'''.

If 1 G2V star is added to 1 M2V star the result may be a double star. The operation of addition here usually requires an explanation (a theory).
Astronomers use math all the time. One way it is used is when we look at objects in the sky with a telescope. The camera that is attached to the telescope basically records a series of numbers - those numbers might correspond to how much light different objects in the sky are emitting, what type of light, etc. In order to be able to understand the information that these numbers contain, we need to use math and statistics to interpret them.
==Inclinations==
{{main|Powers/Tendencies/Inclinations|Inclinations}}
[[Image:Orbit1.svg|thumb|right|200px|The diagram describes the parameters associated with orbital inclination (''i''). Credit: [[w:User:Lasunncty|Lasunncty]].]]
'''Def.''' the angle of intersection of a reference plane is called an '''inclination'''.

"The orbital inclination [''i''] [of Mercury] varies between 5° and 10° with a 10<sup>6</sup> yr period with smaller amplitude variations with a period of about 10<sup>5</sup> yr."<ref name=Peale/>
{{clear}}

==Dimensional analyses==
{{main|Dimensional analyses}}
'''Def.''' a single aspect of a given thing is called a '''dimension'''.

Usually, in astronomy, a number is associated with a dimension or aspect of an entity. For example, the Earth is 1.50 x 10<sup>8</sup> km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 10<sup>8</sup> is a number.

'''Def.''' the study of the dimensions of quantities; used to obtain information about large complex systems, and as a means of checking equations is called '''dimensional analysis'''.

Prefixed values cannot be multiplied or divided together, and they have to be converted into non-prefixed standard form for such calculations. For example, 5&nbsp;mV × 5&nbsp;mA ≠ 25&nbsp;mW. The correct calculation is: 5&nbsp;mV × 5&nbsp;mA = 5 × 10<sup>−3</sup>&nbsp;V × 5 × 10<sup>−3</sup>&nbsp;A = 25 x 10<sup>−6</sup>&nbsp;W = 25&nbsp;µW = 0.025&nbsp;mW.

Prefixes corresponding to an exponent that is divisible by three are often recommended. Hence "100&nbsp;m" rather than "1&nbsp;hm" (hectometre) or "10&nbsp;dam" (decametres).  The "non-three" prefixes (hecto-, deca-, deci-, and centi-) are however more commonly used for everyday purposes than in science.

When units occur in exponentiation, for example, in square and cubic forms, any size prefix is considered part of the unit, and thus included in the exponentiation.
* {{gaps|1|km<sup>2</sup>}} means one square kilometre or the size of a square of 1000&nbsp;m by 1000&nbsp;m and not 1000 [[w:square metre|square metre]]s.
* {{gaps|2|Mm<sup>3</sup>}} means two cubic megametre or the size of two cubes of {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} by {{gaps|1|000|000|m}} or {{gaps|2|e=18|u=m<sup>3</sup>}}, and not {{gaps|2|000|000|cubic metres}} ({{gaps|2|e=6|u=m<sup>3</sup>}}).

;Examples
* {{gaps|5|u=cm}} = {{gaps|5|e=-2|u=m}} = {{gaps|5|&times;|0.01|m}} = {{gaps|0.05|m}}
* {{gaps|3|u=MW}} = {{gaps|3|e=6|u=W}} = {{gaps|3|&times;|1|000|000|W}} = {{gaps|3|000|000|W}}

==Astronomical units==
{{main|Astronomy/Units|Astronomical units}}
'''Def.''' "1 day (d)" is called the '''astronomical unit of time'''.<ref name=Seidelmann>{{ cite book
|author=P. K. Seidelmann
|title=Measuring the Universe The IAU and astronomical units
|publisher=International Astronomical Union
|date=1976
|url=http://www.iau.org/public/measuring/
|accessdate=2011-11-27 }}</ref>

'''Def.''' "the distance from the centre of the Sun at which a particle of negligible mass, in an unperturbed circular orbit, would have an orbital period of 365.2568983 days" is called the '''Astronomical Unit''' (AU).<ref name=Seidelmann/>

'''Def.''' "the distance at which one Astronomical Unit subtends an angle of one arcsecond" is called the '''parsec''' (pc).<ref name=Seidelmann/>

'''Def.''' "365.25 days" is called a '''Julian year'''.<ref name=Seidelmann/>

'''Def.''' "36,525 days" is called a '''Julian century'''.<ref name=Seidelmann/>

{| class="wikitable" style="float:center"
|+ Units of Physics and Astronomy
|-
! Dimension !! Astronomy !! Symbol !! Physics !! Symbol !! Conversion 
|-
| time || 1 day || d || 1 second || s || 1 d = 86,400 s<ref name=Seidelmann/>
|-
| time || 1 "Julian year"<ref name=Wilkins>International Astronomical Union "[http://www.iau.org/science/publications/proceedings_rules/units/ SI units]" accessed February 18, 2010. (See Table 5 and section 5.15.) Reprinted from George A. Wilkins & IAU Commission 5, [http://www.iau.org/static/publications/stylemanual1989.pdf "The IAU Style Manual (1989)"] (PDF file) in ''IAU Transactions'' Vol. XXB</ref> || J || 1 second || s || 1 J = 31,557,600 s 
|-
| distance || 1 astronomical unit || AU || 1 meter || m || 1 AU = 149,597,870.691 km<ref name=Seidelmann/> 
|-
| mass || 1 Sun || M<sub>ʘ</sub> || 1 kilogram || kg || 1 M<sub>ʘ</sub> = 1.9891 x 10<sup>30</sup> kg<ref name=Seidelmann/>
|-
| luminosity || 1 Sun || L<sub>ʘ</sub> || 1 watt || W || 1 L<sub>ʘ</sub> = 3.846 x 10<sup>26</sup> W<ref name=Williams2004>{{ cite book
|author=David R. Williams
|title=Sun Fact Sheet
|publisher=NASA Goddard Space Flight Center
|location=Greenbelt, MD
|date=September 2004
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
|accessdate=2011-12-20 }}</ref>
|-
| angular distance || 1 parsec || pc || 1 meter || m || 1 pc ~ 30.857 x 10<sup>12</sup> km<ref name=Seidelmann/>
|-
|}

==Regions==
{{main|Spaces/Regions|Regions}}
[[Image:Center of the Milky Way Galaxy IV – Composite.jpg|thumb|right|250px|This is a composite image of the central region of our Milky Way galaxy. Credit: NASA/JPL-Caltech/ESA/CXC/STScI.]]
A '''region''' is a division or part of a space having definable characteristics but not always fixed boundaries.

'''Def.''' any [[wikt:considerable|considerable]] and [[wikt:connected|connected]] [[wikt:part|part]] of a [[wikt:space|space]] or [[wikt:surface|surface]]; specifically, a [[wikt:tract|tract]] of [[wikt:land|land]] or [[wikt:sea|sea]] of considerable but [[wikt:indefinite|indefinite]] [[wikt:extent|extent]]; a [[wikt:country|country]]; a [[wikt:district|district]]; in a broad sense, a place without special [[wikt:reference|reference]] to [[wikt:location|location]] or extent but viewed as an [[wikt:entity|entity]] for [[wikt:geographical|geographical]], [[wikt:social|social]] or [[wikt:cultural|cultural]] reasons is called a '''region'''.

'''Region''' is most commonly found as a term used in terrestrial and astrophysics sciences also an area, notably among the different sub-disciplines of [[geography]], studied by [[w:Regional geography|regional geographers]]. Regions consist of [[w:subregions|subregions]] that contain clusters of like areas that are distinctive by their uniformity of description based on a range of statistical data, for example demographic, and locales.

'''Def.''' a subset of <math>\R^n</math> that is [[w:open set|open]] (in the standard [[w:Euclidean topology|Euclidean topology]]), [[w:connected set|connected]] and [[w:empty set|non-empty]] is called a '''region''', or '''region of <math>\R^n</math>, where <math>\R^n</math> is the n-dimensional real number system.'''
{{clear}}

==Areas==
{{main|Spaces/Areas|Areas}}
[[Image:RandLintegrals.png|thumb|right|250px|This diagram shows an approximation to an area under a curve. Credit: [[commons:User:Dubhe|Dubhe]].]]
In the figure on the right, an area is the difference in the x-direction times the difference in the y-direction.

This rectangle cornered at the origin of the curvature represents an area for the curve.
{{clear}}

==Orbits==
{{main|Spaces/Regions/Spheres/Orbits|Orbits}}
[[Image:Kepler orbits.svg|thumb|right|250px|A hyperbolic pass is indicated by the blue line with an eccentricity of 1.3. A parabolic pass is the green line. The elliptical orbit in red has an eccentricity (''e'') of 0.7. Credit: [[commons:User:Stamcose|Stamcose]].]]
[[Image:Orbit3.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with circular orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:Orbit5.gif|thumb|right|250px|Two bodies orbiting around a common barycenter (red cross) with elliptic orbits. Credit: [[commons:User:Zhatt|Zhatt]].]]
[[Image:ISEE3-ICE-trajectory.gif|thumb|left|250px|ISEE-3 is inserted into a "halo" orbit on June 10, 1982. Credit: NASA.]]
'''Def.''' a circular or elliptical path of one object around another object is called an '''orbit'''.

Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of [[w:epicycles|epicycles]], which are the sums of numerous circular motions.<ref>''Encyclopaedia Britannica'', 1968, vol. 2, p. 645</ref> Theories of this kind predicted paths of the planets moderately well, until [[w:Johannes Kepler|Johannes Kepler]] was able to show that the motions of planets were in fact (at least approximately) elliptical motions.<ref name=Caspar>M Caspar, ''Kepler'' (1959, Abelard-Schuman), at pp.131–140; A Koyré, ''The Astronomical Revolution: Copernicus, Kepler, Borelli'' (1973, Methuen), pp. 277–279</ref>

In the [[w:geocentric model|geocentric model]] of the solar system, the [[w:celestial spheres|celestial spheres]] model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as [[w:deferent and epicycle|deferent and epicycle]]s were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.

In [[Astronomy/Theory|theoretical astronomy]], whether the Earth moves or not, serving as a fixed point with which to measure movements by objects or entities, or there is a [[w:solar system|solar system]] with the [[Stars/Sun|Sun]] near its center, is a matter of simplicity and calculational accuracy. Copernicus's theory provided a strikingly simple explanation for the apparent retrograde motions of the planets&mdash;namely as [[w:parallax|parallactic]] displacements resulting from the Earth's motion around the Sun&mdash;an important consideration in [[w:Johannes Kepler|Johannes Kepler]]'s conviction that the theory was substantially correct.<ref name=Linton>{{ cite book
|author=Christopher M. Linton
|title=From Eudoxus to Einstein&mdash;A History of Mathematical Astronomy
|publisher=Cambridge University Press
|location=Cambridge
|date=2004
|editor=
|pages=
|url=
|bibcode=
|doi=
|pmid=
|isbn=978-0-521-82750-8
}}</ref> "[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy"<ref name=Linton/> with the Earth at the center. Using a computer, this means that for competing programs, one written for each theory, the heliocentric program finishes first (for a mutually specified high degree of accuracy).

Orbits come in many shapes and motions. The simplest forms are a circle or an ellipse.
{{clear}}

==Infinitesimals==
{{main|Infinitesimals}}
'''Notation''': let the symbol <math>d</math> represent an '''infinitesimal difference in'''.

'''Notation''': let the symbol <math>\partial</math> represent an '''infinitesimal difference in''' one of more than one.

==Distances==
{{main|Distances}}
[[Image:Distancedisplacement.svg|thumb|right|250px|Distance along a path is compared in this diagram with displacement. Credit: .]]
'''Def.''' the amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line is called a '''distance'''.

'''Distance''' (or '''farness''') is a numerical description of how far apart objects are.  In [[physics]] or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over").  In [[mathematics]], a distance function or [[w:Metric (mathematics)|metric]] is a generalization of the concept of physical distance.  A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other.

'''Def.'''
# a series of interconnected rings or links usually made of metal,
# a series of interconnected links of known length, used as a measuring device,
# a long measuring tape,
# a unit of length equal to 22 yards. The length of a Gunter's surveying chain. The length of a cricket pitch. Equal to 20.12 metres. Equal to 4 rods. Equal to 100 links.
# a totally ordered set, especially a totally ordered subset of a poset,
# iron links bolted to the side of a vessel to bold the dead-eyes connected with the shrouds; also, the channels, or
# the warp threads of a web

is called a '''chain'''.

'''Def.''' a unit of length equal to 220 yards or exactly 201.168 meters, now only used in measuring distances in horse racing is called a '''furlong'''.

'''Def.'''
# a trench cut in the soil, as when plowed in order to plant a crop or
# any trench, channel, or groove, as in wood or metal

is called a '''furrow'''.

'''Def.''' the distance that a person can walk in one hour, commonly taken to be approximately three English miles (about five kilometers) is called a '''league'''.

:<math>R_{\odot} (equatorial) = 696,342 km</math>

:<math>R_J (equatorial) = 71,492 km</math>

:<math>R_S (equatorial) = 60,268 km</math>

:<math>R_U (equatorial) = 25,559 km</math>

:<math>R_N (equatorial) = 24,764 km</math>

Then,

:<math>R_J (equatorial) = F_J * R_{\odot} (equatorial),</math>
:<math>R_S (equatorial) = F_S * R_{\odot} (equatorial),</math>
:<math>R_U (equatorial) = F_U * R_{\odot} (equatorial), and</math>
:<math>R_N (equatorial) = F_N * R_{\odot} (equatorial).</math>
{{clear}}

==Cosmic distance ladders==
{{main|Distances/Extragalactics|Cosmic distance ladders}}
The [[w:apparent magnitude|apparent magnitude]], or the magnitude as seen by the observer, can be used to determine the distance ''D'' to the object in kiloparsecs (where 1&nbsp;kpc equals 1000&nbsp;parsecs) as follows:

:<math>\begin{smallmatrix}5 \cdot \log_{10} \frac{D}{\mathrm{kpc}}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}</math>

where ''m'' the apparent magnitude and ''M'' the absolute magnitude.

==Diameters==
{{main|Dimensions/Breadths/Diameters|Diameters}}
'''Def.''' the length of any straight line between two points on the circumference of a circle that passes through the centre/center of the circle is called a '''diameter'''.

==Arithmetic dimensional analysis==

Usually, pure [[arithmetic]] only involves numbers. But, when arithmetic is used in a science such as [[radiation astronomy]], dimensional analysis is also applicable.

To build an observatory usually requires adding components together.

For example: 1 dome + 1 telescope + 1 outbuilding + 1 control room + 1 laboratory + 1 observation room may = 1 observatory.

Yet,

: 1 + 1 + 1 + 1 + 1 + 1 = 6 components in 1 simple observatory.

==Obliquities==
{{main|Spaces/Obliquities|Obliquities}}
'''Def.''' the quality of being oblique in direction, deviating from the horizontal or vertical; or the angle created by such a deviation is called '''obliquity'''.

'''Axial tilt''' (also called '''obliquity''') is the angle between an object's [[w:Axis of rotation|rotational axis]], and a line [[w:Perpendicular|perpendicular]] to its [[w:Orbital plane (astronomy)|orbital plane]]. The planet [[w:Venus|Venus]] has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like [[Earth]]. The planet [[w:Uranus|Uranus]] is rotating on its side in such a way that its rotational axis, and hence its north pole, is pointed almost in the direction of its orbit around the [[Stars/Sun|Sun]]. Hence the axial tilt of Uranus is 97°.<ref name=Williams>{{ cite book
|author=David R. Williams
|title=Planetary Fact Sheet Notes
|url=http://nssdc.gsfc.nasa.gov/planetary/factsheet/planetfact_notes.html }}</ref>

The obliquity of the Earth's axis has a period of about 41,000 years.<ref name=Hays>{{ cite journal
|author=J. D. Hays
|author2=John Imbrie
|author3=N. J. Shackleton
|title=Variations in the Earth's Orbit: Pacemaker of the Ice Ages
|journal=Science
|month=December
|year=1976
|volume=194
|issue=4270
|pages=
|url=http://www.whoi.edu/science/GG/paleoseminar/ps/hays76.ps
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2011-11-08 }}</ref>

==Inverses==
{{main|Inverses}}
'''Def.''' the set of points that map to a given point (or set of points) under a specified function is called an '''inverse image'''.

Under the function given by <math>f(x)=x^2</math>, the '''inverse image''' of 4 is <math>\{-2,2\}</math>, as is the '''inverse image''' of <math>\{4\}</math>.

==Antapex==

'''Def.''' the point to which the Sun appears to be moving with respect to the local stars is called the '''solar apex'''.

An antapex is a point that an astronomical object's total motion is directed away from. It is opposite to the apex.

The '''local standard of rest''' or '''LSR'''  follows the mean motion of material in the [[w:Milky Way|Milky Way]] in the neighborhood of the [[Stars/Sun|Sun]].<ref name= Shu>{{cite book
|title=The Physical Universe
|author= Frank H Shu
|page= 261
|url=http://books.google.com/?id=v_6PbAfapSAC&pg=PA261
|isbn=0935702059
|date=1982
|publisher=University Science Books }}</ref> The path of this material is not precisely circular.<ref name=Binney>{{cite book
|title=Galactic Astronomy
|author=James Binney
|author2=Michael Merrifield
|url=http://books.google.com/?id=arYYRoYjKacC&pg=PA536
|page= 536
|date=1998
|isbn=0691025657
|publisher=Princeton University Press }}</ref> The Sun follows the '''solar circle''' ([[w:Ellipse#Eccentricity|eccentricity]] ''e'' < 0.1 ) at a speed of about 220&nbsp;km/s in a clockwise direction when viewed from the [[w:galactic coordinates|galactic north pole]] at a radius of ≈ 8 [[w:kiloparsec|kpc]] about the center of the galaxy near [[w:Sgr A*|Sgr A*]], and has only a slight  motion, towards the [[w:Solar apex|solar apex]], relative to the LSR.<ref name=Reid>{{ cite book
|url=http://books.google.com/?id=bP9hZqoIfhMC&pg=PA19
|title=Mapping the Galaxy and Nearby Galaxies
|editor=F. Combes, Keiichi Wada
|date=2008
|publisher=Springer
|isbn=0387727671
|author= Mark Reid
|display-authors=etal
|chapter=Mapping the Milky Way and the Local Group
|pages=19–20 }}</ref> The Sun's [[w:peculiar motion|peculiar motion]] relative to the LSR is 13.4 km/s.<ref name=Binney1>{{ cite book
|author=Binney J.
|author2=Merrifield M.
|title=op. cit.
|isbn=0691025657
|chapter=§10.6
}}</ref><ref name=Mamajek>{{cite journal
|title=On the distance to the Ophiuchus star-forming region
|journal=Astron. Nachr.
|volume=AN 329
|doi= 10.1002/asna.200710827
|arxiv=0709.0505
|year=2008
|page=12
|author=E.E. Mamajek
|bibcode = 2008AN....329...10M }}</ref> The LSR velocity is anywhere from 202–241&nbsp;km/s.<ref name=Majewski>{{ cite journal
|title=Precision Astrometry, Galactic Mergers, Halo Substructure and Local Dark Matter
|author=Steven R. Majewski
|journal=Proceedings of IAU Symposium 248
|arxiv=0801.4927
|year=2008
|bibcode = 2008IAUS..248..450M
|doi = 10.1017/S1743921308019790
|volume=3 }}</ref>

==Algebras==
{{main|Algebras}}
'''Notation''': let the symbol '''*''' designate an as yet unspecified operation.

'''Notation''': let the symbol '''R''' designate an as yet unspecified relation.

'''Def.''' a system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols is called an '''algebra'''.

Fundamentally, [[Portal:Algebra|algebra]] uses letters to represent as yet unspecified [[numbers]]. The numbers may be [[Numbers/Integers|integers]], [[The Number System#Rational Numbers|rational numbers]], [[The Number System#Irrational Numbers|irrational numbers]], or any [[Real Numbers|real number]] or [[Complex Numbers|complex number]]. As an [[w:Experimentalist|experimentalist]], eventually you must find a way to change unspecified numbers into specified ones. But, as a theoretician, first you are free to leave the numbers in some algebraic form, then to have your theory tested by any experimentalist you need to relate the algebraic terms of your theory to real or complex numbers.

Consider the lower case letters of the English alphabet: a and n. The statement, "a * n R an", contains the operation * (followed by) and the relation R (spells the word).

The manipulations of these symbols are performed using operations.

'''Def.''' a [[wikt:procedure|procedure]] for generating a [[wikt:value|value]] from one or more other values (the [[wikt:operand|operand]]s; the value for any particular [operand] is unique) is called an '''operation'''.

'''Notation''': let the symbol <math>\sum</math> represent the summation of many terms.

'''Notation''': let the symbol <math>\Pi</math> represent the product of many terms.

The results are recorded using statements of relation.

'''Def.''' a relation in which each element of the domain is associated with exactly one element of the codomain is called a '''function'''.

==Geometries==
{{main|Geometries}}
[[Image:Similar-geometric-shapes.svg|thumb|right|250px|Mathematics: Figures shown in the same color are similar. Credit: [[commons:User:Amada44|Amada44]].]]
'''Def.''' of geometrical figures including triangles, squares, ellipses, arcs and more complex figures, having the same shape but possibly different size, rotational orientation, and position; in particular, having corresponding angles equal and corresponding line segments proportional; such that one can be had from the other using a sequence of operations of rotation, translation and scaling is called '''similar'''.

'''Def.''' a branch of mathematics that studies solutions of systems of algebraic equations using both algebra and geometry is called '''algebraic geometry'''.

'''Def.''' a branch of mathematics that investigates properties of figures through the coordinates of their points is called '''analytic geometry'''.

'''Def.''' a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane is called '''projective geometry'''.

'''Def.''' a set along with a collection of finitary functions and relations is called a '''structure'''.

'''Def.'''
# a set of points with some added structure,
# distance between things,
# physical extent a range of values or locations across two or three dimensions,
# physical extent in all directions, seen as an attribute of the universe,
# a set of points, each of which is uniquely specified by a number (the dimensionality) of coordinates,
# a generalized construct or set whose members have some property in common; typically there will be a geometric metaphor allowing these members to be viewed as "points",
# a gap; an empty place,
# a (chiefly empty) area or volume with set limits or boundaries,

is called a '''space'''.

The universe as often perceived may be described spatially, sometimes with plane [[geometry]], other occasions with [[Ideas in Geometry/Spherical Geometry|spherical geometry]].
{{clear}}

==Coordinates==
{{main|Measurements/Coordinates|Coordinates}}
[[Image:Cartesian-coordinate-system-with-circle.svg|thumb|right|250px|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (''x'' - ''a'')<sup>2</sup> + (''y'' - ''b'')<sup>2</sup> = ''r''<sup>2</sup> where ''a'' and ''b'' are the coordinates of the center (''a'', ''b'') and ''r'' is the radius. Credit: [[w:User:345Kai|345Kai]].]]
A '''Cartesian coordinate system''' specifies each point uniquely in a plane by a pair of numerical '''coordinates''', which are the [positive and negative numbers] signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a ''coordinate axis'' or just ''axis'' of the system, and the point where they meet is its ''origin'', usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
{{clear}}

===Triclinic coordinates===
[[Image:Reseaux 3D aP.png|thumb|right|100px]] A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol "⋀" means "angle between".
{{clear}}

===Monoclinic coordinates===
[[Image:Monoclinic cell.svg|thumb|right|100px]] In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°.
{{clear}}

===Orthorhombic coordinates===
[[Image:Reseaux 3D oP.png|thumb|right|100px]] In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c.
{{clear}}

===Tetragonal coordinates===
[[Image:Reseaux 3D tP-2011-03-12.png|thumb|right|100px]] A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c.
{{clear}}

===Rhombohedral coordinates===
[[Image:Rhombohedral.svg|thumb|right|100px]] A rhombohedral system has a = b = c and α = β = γ < 120°, ≠ 90°.
{{clear}}

===Hexagonal coordinates===
[[Image:Reseaux 3D hP.png|thumb|right|100px]] A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°.
{{clear}}

==Triangles==
{{main|Spaces/Angularities/Triangles|Triangles}}
[[Image:Simple triangle.svg|thumb|right|200px|This diagram shows a regular '''triangle''', the geometric shape. Credit: [[commons:User:Dbc334|Dbc334]].]]
'''Def.''' a polygon with three sides and three angles is called a '''triangle'''.
{{clear}}

==Curvatures==
{{main|Spaces/Curvatures|Curvatures}}
The graph at the top of [[Astronomy/Mathematics#Areas|areas]] shows a curve or curvature.

==Conic sections==
{{main|Forms/Rotundity/Conics/Sections|Conic sections}}
[[Image:Conic Sections.svg|thumb|right|250px|alt=Diagram of conic sections|Conics are of three types: parabolas , ellipses, including circles, and hyperbolas. Credit: .]]
[[Image:Ellipse parameters 2.svg|left|thumb|300px|Conic parameters are shown in the case of an ellipse. Credit: .]]
'''Def.''' any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola is called a '''conic section'''.

In [[mathematics]], a '''conic section''' (or just '''conic''') is a [[w:curve|curve]] obtained as the intersection of a [[w:cone (geometry)|cone]] (more precisely, a right circular [[w:conical surface|conical surface]]) with a [[w:plane (mathematics)|plane]].

Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of ''a''>''b'', for which the major axis is horizontal; for the reverse case, interchange the symbols ''a'' and ''b''. For the hyperbola the east-west opening case is given. In all cases, ''a'' and ''b'' are positive.)

{| class="wikitable"
! conic section
! equation
! eccentricity (''e'')
! linear eccentricity (''c'')
! semi-latus rectum (''ℓ'')
! focal parameter (''p'')
|-
|  [[w:circle|circle]]
|| <math>x^2+y^2=a^2 \,</math>
|| <math> 0 \,</math>
|| <math> 0 \,</math>
|| <math> a \,</math>
|| <math> \infty</math>
|-
|  [[w:ellipse|ellipse]]
|| <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>
|| <math>\sqrt{1-\frac{b^2}{a^2}}</math>
|| <math>\sqrt{a^2-b^2}</math>
|| <math>\frac{b^2}{a}</math>
|| <math>\frac{b^2}{\sqrt{a^2-b^2}}</math>
|-
|  [[w:parabola|parabola]]
|| <math>y^2=4ax \,</math>
|| <math> 1 \,</math>
|| <math> a \, </math>
|| <math> 2a \,</math>
|| <math> 2a \, </math>
|-
|  [[Conic sections|hyperbola]]
|| <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1</math>
|| <math>\sqrt{1+\frac{b^2}{a^2}}</math>
|| <math>\sqrt{a^2+b^2}</math>
|| <math>\frac{b^2}{a}</math>
|| <math>\frac{b^2}{\sqrt{a^2+b^2}}</math>
|}

The general parabola equation with a vertical axis

:<math>ax^2+bx+c=0</math>

is solved in terms of the constants ''a'', ''b'', and ''c'' for x by

:<math>x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.</math>

The general conic equation in a Cartesian plane (x,y) is

:<math> A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,,</math>

where

:<math> (AC - B^2/4)F + BED/4 - CD^2/4 - AE^2/4 \ne 0.</math>

For parabolas,

:<math> B^2 = 4AC,</math>

where ''A'' and ''C'' are not both zero.
{{clear}}

==Variations==
{{main|Spaces/Variations|Variations}}
'''Def.''' a partial change in the form, position, state, or qualities of a thing or a related but distinct thing is called a '''variation'''.

==Precessions==
{{main|Motions/Precessions|Precessions}}
'''Def.''' any of several slow changes in an astronomical body's rotational or orbital parameters such as the slow gyration of the [[Earth]]’s axis around the pole of the ecliptic is called a '''precession'''.

'''Def.''' the slow westward shift of the equinoxes along the plane of the ecliptic, resulting from precession of an object's axis of rotation, and causing the equinoxes to occur earlier each year is called the '''precession of the equinoxes'''.

The equinoxes of Earth precess with a period of about 21,000 years.<ref name=Hays/>

==Rotations==
{{main|Rotations}}
[[Image:Rotating Sphere.gif|right|thumb|250px|A [[w:polyhedron|polyhedron]] resembling a sphere rotating around an axis. Credit: [[w:User:BorisFromStockdale|BorisFromStockdale]].]]
'''Def.''' the act of turning around a centre or an axis is called a '''rotation'''.

A '''rotation''' is a [[w:circular motion|circular]] movement of an object around a ''center'' (or ''[[w:point (geometry)|point]]'') ''of rotation''. A [[w:Three-dimensional space|three-dimensional]] object rotates always around an imaginary [[w:Line (geometry)|line]] called a ''rotation axis''. If the axis is within the body, and passes through its [[w:center of mass|center of mass]] the body is said to rotate upon itself, or ''[[wikt:spin|spin]]''. A '''rotation''' about an external point, e.g. the [[Earth]] about the [[Stars/Sun|Sun]], is called a revolution or ''[[w:orbit|orbital revolution]]''.

Axes of rotation can be multiple:
# one-fold - ⨀, ⦺, ⧀
# two-fold - ⨸,
# three-fold - ▲,
# four-fold - ◈,
# five-fold - ✪, or
# six-fold - ✱.

Higher-fold axes of rotation are possible. As the number-fold of axes increases, the polyhedron approaches a circle. Or, in three dimensions, a sphere.
{{clear}}

==Mirror planes==
{{main|Mirror planes}}
A mirror plane reflects on the other side the handedness that is on the initial side:
# ⨴ | ⨵, the plane between is the mirror so that on either side is the reflection of the other, here an axis of rotation out of the plane of the paper could place the reflection on top of the object on the other side of the mirror plane,
# ∀ | ∀, here such an axis of rotation would not work,
# ⊆ | ⊇, this one is like number two,
# ⊕ | ⊕, here rotational symmetry is preserved, and
# ⨫ | ⨬, here rotation axes exist in the plane of the paper.

==Resonances==
{{main|Resonances}}
[[Image:Galilean moon Laplace resonance animation.gif|right|thumb|365px|The [[w:Laplace resonance|Laplace resonance]]s of Ganymede, [[Rocks/Ice sheets/Europa|Europa]], and [[Io]] is illustrated. Credit: User:Matma Rex.]]
An orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of [[Jupiter]]'s moons [[Rocks/Rocky objects/Ganymede|Ganymede]], [[Rocks/Ice sheets/Europa|Europa]] and [[Io]], and the 2:3 resonance between [[Pluto]] and [[Neptune]]. Unstable resonances with [[Saturn]]'s inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large [[Solar System]] bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.
{{clear}}

==Eccentricities==
{{main|Eccentricities}}
'''Def.''' the ratio, constant for any particular conic section, of the distance of a point from the focus to its distance from the [[wikt:directrix|directrix]] is called the '''eccentricity'''.

For an ellipse, the eccentricity is the ratio of the distance from the center to a focus divided by the length of the semi-major axis.

"Mercury's orbit eccentricity [''e''] varies between about 0.11 and 0.24 with the shortest time lapse between the extremes being about 4 x 10<sup>5</sup> yr".<ref name=Peale>{{ cite journal
|author=Peale, S. J.
|title=Possible histories of the obliquity of Mercury
|journal=Astronomical Journal
|month=June
|year=1974
|volume=79
|issue=6
|pages=722-44
|bibcode=1974AJ.....79..722P
|doi=10.1086/111604
|pmid= }}</ref> "Smaller amplitude variations occur with about a 10<sup>5</sup> yr period."<ref name=Peale/>

==Spherical geometries==
{{main|Geometries/Spheres|Spherical geometries}}
[[Image:Triangles (spherical geometry).jpg|thumb|right|250px|On a sphere, the sum of the angles of a triangle is not equal to 180°. Credit: .]]
'''Def.''' the [[wikt:non-Euclidean geometry|non-Euclidean geometry]] on the surface of a [[wikt:sphere|sphere]] is called '''spherical geometry'''.

'''Spherical geometry''' is the [[geometry]] of the two-[[w:dimension|dimension]]al surface of a [[sphere]]. It is an example of a [[geometry]] which is not Euclidean. Two practical applications of the principles of spherical geometry are to [[w:navigation|navigation]] and astronomy.

A sphere [suggested by the image of the Earth at right] is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional [[w:manifold|manifold]].

The '''great-circle''' or '''[[w:Great Circle|orthodromic]] distance''' is the shortest [[w:distance|distance]] between any two [[w:Point (geometry)|point]]s on the surface of a [[w:sphere|sphere]] measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is different from ordinary [[w:Euclidean geometry|Euclidean geometry]], the equations for distance take on a different form. The distance between two points in [[w:Euclidean space|Euclidean space]] is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In [[w:non-Euclidean geometry|non-Euclidean geometry]], straight lines are replaced with [[w:geodesic|geodesic]]s. Geodesics on the sphere are the ''[[w:great circle|great circle]]s'' (circles on the sphere whose centers are coincident with the center of the sphere).

Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the [[w:Riemannian circle|Riemannian circle]].

==Logical laws==
{{main|Maxims/Axioms/Logical laws|Logical laws}}
[[Image:Kepler laws diagram.svg|thumb|300px|The diagram illustrates Kepler's three laws using two planetary orbits. Credit: [[commons:User:Hankwang|Hankwang]].]]
Kepler's laws of planetary motion:
# The orbit of every planet is an [[w:ellipse|ellipse]] with the Sun at one of the two [[w:Focus (geometry)|foci]].
# A [[w:line (geometry)|line]] joining a planet and the Sun sweeps out equal [[w:area|area]]s during equal intervals of time.<ref name="Wolfram2nd">Bryant, Jeff; Pavlyk, Oleksandr. "[http://demonstrations.wolfram.com/KeplersSecondLaw/ Kepler's Second Law]", ''Wolfram Demonstrations Project''. Retrieved December 27, 2009.</ref>
# The [[w:square (algebra)|square]] of the [[w:orbital period|orbital period]] of a planet is directly [[w:Proportionality (mathematics)|proportional]] to the [[w:cube (arithmetic)|cube]] of the [[w:semi-major axis|semi-major axis]] of its orbit.

The diagram at the right illustrates Kepler's three laws of planetary orbits: (1) The orbits are ellipses, with focal points ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>2</sub> for the first planet and ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>3</sub> for the second planet. The Sun is placed in focal point ''&fnof;''<sub>1</sub>. (2) The two shaded sectors ''A''<sub>1</sub> and ''A''<sub>2</sub> have the same surface area and the time for planet 1 to cover segment ''A''<sub>1</sub> is equal to the time to cover segment ''A''<sub>2</sub>. (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''<sub>1</sub><sup>3/2</sup>&nbsp;:&nbsp;''a''<sub>2</sub><sup>3/2</sup>.

The simplest description of the paths astronomical objects may take when passing each other includes a hyperbolic and parabolic pass. When capture occurs it usually produces an elliptical orbit.

==Horizontal coordinate system==
{{main|Coordinates/Horizontals|Horizontal coordinate systems}}
[[Image:Horizontal coordinate system 2.svg|thumb|right|250px|This diagram describes altitude and azimuth. Credit: Francisco Javier Blanco González.]]
The altitude of an entity in the sky is given by the angle of the arc from the local horizon to the entity.

The horizontal coordinate system is a [[w:celestial coordinate system|celestial coordinate system]] that uses the observer's local [[w:horizon|horizon]] as the [[w:Fundamental plane (spherical coordinates)|fundamental plane]]. This coordinate system divides the sky into the upper [[w:sphere|hemisphere]] where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in the way. The [[w:Great circle|great circle]] separating hemispheres [is] called [the] celestial horizon or rational horizon. The pole of the upper hemisphere is called the [[w:Zenith|zenith]]. The pole of the lower hemisphere is called the [[w:Nadir|nadir]].<ref name=Schombert>{{ cite book
|url=http://abyss.uoregon.edu/~js/ast121/lectures/lec03.html
|title=Earth Coordinate System
|author=James Schombert
|publisher=University of Oregon Department of Physics
|accessdate=19 March 2011 }}</ref>

The horizontal coordinates are:
* '''Altitude (Alt)''', sometimes referred to as [[w:elevation (disambiguation) | elevation]],  is the angle between the object and the observer's local horizon. It is expressed as an angle between 0 degrees to 90 degrees.
* '''[[w:Azimuth|Azimuth]] (Az)''', that is the angle of the object around the horizon, usually measured from the north increasing towards the east. 
* '''Zenith distance''', the distance from directly overhead (i.e. the zenith) is sometimes used instead of altitude in some calculations using these coordinates.  The zenith distance is the [[w:complementary angles|complement]] of altitude (i.e. 90°-altitude).

==Fixed point in the sky==
[[Image:EquatorialDecRA.png|thumb|100px|right|By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA). Credit: .]]
[[Image:AxialTiltObliquity.png|thumb|250px|right|Earth is shown as viewed from the Sun; the orbit direction is counter-clockwise (to the left). Description of the relations between axial tilt (or obliquity), rotation axis, plane of orbit, celestial equator and ecliptic. Credit: .]]
The observations require precise [[w:measurement|measurement]] and adaptations to the movements of the Earth, especially when and where, for a time, an object or entity is available.

With the creation of a geographical grid, an observer needs to be able to fix a point in the sky. From many observations within a period of stability, an observer notices that patterns of visual objects or entities in the night sky repeat. Further, a choice is available: is the Earth moving or are the star patterns moving? Depending on latitude, the observer may have noticed that the days vary in length and the pattern of variation repeats after some number of days and nights. By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure [[w:equatorial coordinates|equatorial coordinates]]: [[w:Declination|declination]] (Dec) and [[w:Right ascension|right ascension]] (RA).

Once these can be determined, the apparent absolute positions of objects or entities are available in a communicable form. The repeat pattern of (day/night)s allows the observer to calculate the RA and Dec at any point during the cycle for a new object, or approximations are made using RA and Dec for recognized objects.

Independent of the choice made (Earth moves or not), the pattern of objects is the same for days or nights of the repeating length once a year. The '''[[w:Equinox|vernal equinox]]''' is a day/night of equal length and the same pattern of objects in the night sky. The '''autumnal equinox''' is the other equal length day/night with its own pattern of objects in the night sky.

The projection of the Earth's equator and poles of rotation, or if the observer hasn't concluded as yet that it's the Earth that's rotating, the circulating pattern of stars in ever smaller circles heading in specific directions, is the celestial sphere.
{{clear}}

==Trigonometries==
{{main|Trigonometries}}
'''Def.''' the relationships between the [[wikt:side|side]]s and the [[wikt:angle|angle]]s of [[wikt:triangle|triangle]]s and the [[wikt:calculation|calculation]]s based on them is called '''trigonometry'''.

'''Trigonometry''' ... studies [[w:triangle|triangle]]s and the relationships between their sides and the angles between these sides. Trigonometry defines the [[w:trigonometric functions|trigonometric functions]], which describe those relationships and have applicability to cyclical phenomena, such as waves.

==Angular displacement==

For the speeds in units of ''c'', ''β'' = ''v''/''c'', "[i]n the usual interpretation of superluminal motion, the apparent velocity is given by
:<math>\beta_{app} = { \beta_{jet} \sin \phi \over 1 - \beta_{jet} \cos \phi },</math>

where ''β''<sub>jet</sub>''c'' is the jet velocity, and the jet makes an angle ''Φ'' to the line of sight."<ref name=Gabuzda>{{ cite journal
|author=D. C. Gabuzda
|author2=J. F. C. Wardle
|author3=D. H. Roberts
|title=Superluminal motion in the BL Lacertae object OJ 287
|journal=The Astrophysical Journal
|month=January 15,
|year=1989
|volume=336
|issue=1
|pages=L59-62
|url=
|arxiv=
|bibcode=1989ApJ...336L..59G
|doi=10.1086/185361
|pmid=
|accessdate=2012-03-21 }}</ref>

==Radius of the Earth==

Because the [[Earth]] is not perfectly spherical, no single value serves as its natural [[wikt:radius|radius]]. ''Earth radius'' is used as a unit of distance, especially in astronomy and [[geology]]. Any radius a distance from a point on the surface to the center falls between the polar minimum of about 6,357&nbsp;km and the equatorial maximum of about 6,378&nbsp;km (≈3,950 – 3,963&nbsp;mi).

Equations for great-circle distance can be used to roughly calculate the shortest distance between points on the surface of the Earth (''as the crow flies''), and so have applications in [[w:navigation|navigation]].

Let <math>\phi_s,\lambda_s;\ \phi_f,\lambda_f\;\!</math> be the geographical [[w:latitude|latitude]] and [[w:longitude|longitude]] of two points (a base "standpoint" and the destination "forepoint"), respectively, and <math>\Delta\phi,\Delta\lambda\;\!</math> their absolute differences; then <math>\Delta\widehat{\sigma}\;\!</math>, the [[w:central angle|central angle]] between them, is given by the [[w:spherical law of cosines|spherical law of cosines]]:

:<math>{\color{white}\Big|}\Delta\widehat{\sigma}=\arccos\big(\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda\big).\;\!</math>

The distance ''d'', i.e. the [[w:arc length|arc length]], for a sphere of radius ''r'' and <math>\Delta \widehat{\sigma}\!</math> given in

:<math>d = r \, \Delta\widehat{\sigma}.\,\!</math>

This arccosine formula above can have large [[w:rounding error|rounding error]]s if the distance is small (if the two points are a kilometer apart the cosine of the central angle comes out 0.99999999). An equation known as the [[w:haversine formula|haversine formula]] is [[w:Condition number|numerically better-conditioned]] for small distances:<ref name=Sinnott>R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159</ref>

A formula that is accurate for all distances is the following special case (a sphere, which is an ellipsoid with equal major and minor axes) of the [[w:Vincenty's formulae|Vincenty formula]] (which more generally is a method to compute distances on ellipsoids):<ref name=Vincenty>{{ cite journal
  | last = Vincenty
  | first = Thaddeus 
  | authorlink = Thaddeus Vincenty
  | title = Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations
  | journal = Survey Review
  | volume = 23
  | issue = 176
  | pages = 88–93
  | publisher = Directorate of Overseas Surveys
  | location = Kingston Road, Tolworth, Surrey
  | date = 1975-04-01
  | url = http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
  | format = PDF
  | accessdate = 2008-07-21 }}</ref>

:<math>{\color{white}\frac{\bigg|}{|}|}\Delta\widehat{\sigma}=\arctan\left(\frac{\sqrt{\left(\cos\phi_f\sin\Delta\lambda\right)^2+\left(\cos\phi_s\sin\phi_f-\sin\phi_s\cos\phi_f\cos\Delta\lambda\right)^2}}{\sin\phi_s\sin\phi_f+\cos\phi_s\cos\phi_f\cos\Delta\lambda}\right).\;\!</math>

When programming a computer, one should use the <code>[[w:atan2|atan2]]()</code> function rather than the ordinary arctangent function (<code>atan()</code>), in order to simplify handling of the case where the denominator is zero, and to compute <math>\Delta\widehat{\sigma}\;\!</math> unambiguously in all quadrants. Also, make sure that all latitudes and longitudes are in radians (rather than degrees) if that is what your programming language's sin(), cos() and atan2() functions expect (1 radian = 180 / &pi; degrees, 1 degree = &pi; / 180 radians).

==Distance computation==
[[Image:Stellarparallax2.svg|thumb|125px|right|The diagram describes [[w:Stellar parallax|Stellar parallax]] motion.]]
Distance measurement by parallax is a special case of the principle of [[w:triangulation|triangulation]], which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 [[w:arcsecond|arcsecond]],<ref name=ZG44>{{harvnb|Zeilik|Gregory|1998 | loc=p. 44}}.</ref> leaving the other two close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Assuming the angle is small (see [[w:Parallax#Derivation|derivation]] below), the distance to an object (measured in [[w:parsec|parsec]]s) is the [[w:Reciprocal (mathematics)|reciprocal]] of the parallax (measured in [[w:arcsecond|arcsecond]]s): <math>d (\mathrm{pc}) = 1 / p (\mathrm{arcsec}).</math> For example, the distance to [[w:Proxima Centauri|Proxima Centauri]] is 1/0.7687=1.3009 parsecs (4.243 ly).<ref name="apj118">{{cite journal
 | author=Benedict
 | title=Interferometric Astrometry of Proxima Centauri and Barnard's Star Using HUBBLE SPACE TELESCOPE Fine Guidance Sensor 3: Detection Limits for Substellar Companions
 | journal=The Astronomical Journal
 | year=1999 | volume=118 | issue=2 | pages=1086–1100
 | bibcode=1999astro.ph..5318B
 | doi=10.1086/300975
 | ref=harv |arxiv = astro-ph/9905318
 | author-separator=,
 | author2=G. Fritz
 | display-authors=2
 | last3=Chappell
 | first3=D. W.
 | last4=Nelan
 | first4=E.
 | last5=Jefferys
 | first5=W. H.
 | last6=Van Altena
 | first6=W.
 | last7=Lee
 | first7=J.
 | last8=Cornell
 | first8=D.
 | last9=Shelus
 | first9=P. J. }}</ref>
{{clear}}

==Distance to the Moon==

Any distance to the Moon is often initially calculated as a multiple of the Earth radius <math>R_\oplus</math>.

==Parallaxes==

'''Parallax''' is a displacement or difference in the [[w:apparent position|apparent position]] of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines."<ref name=Shorter>{{ cite book
| quote=Mutual inclination of two lines meeting in an angle
| title=Shorter Oxford English Dictionary
| date=1968 }}</ref>"''Astron.'' Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view, and constituting a measure of the distance of the object."<ref name=Oxford>{{ cite book
| title=Oxford English Dictionary
| date=1989
| edition=Second
| quote=
| url=http://dictionary.oed.com/cgi/entry/50171114?single=1&query_type=word&queryword=parallax&first=1&max_to_show=10 }}</ref>

Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to celestial objects including to the [[Moon]], the [[Sun]], and to [[w:star|star]]s beyond the [[Solar System]].

==Diurnal parallax==

''Diurnal parallax'' is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the [[w:terrestrial planet|terrestrial planet]]s or [[w:asteroid|asteroid]]s seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars."<ref name=Seidelmann2005>{{ cite book
| author=P. Kenneth Seidelmann
| date=2005
| title=Explanatory Supplement to the Astronomical Almanac
| publisher=University Science Books
| pages=123–125
| isbn=1891389459 }}</ref><ref name=Barbieri>{{ cite book
| author=Cesare Barbieri
| date=2007
| title=Fundamentals of astronomy
| pages=132–135
| publisher=CRC Press
| isbn=0750308869 }}</ref>

==Lunar parallax==
[[Image:Lunaparallax.png|thumb|250px|right|Diagram of daily lunar parallax. Credit: .]]
[[Image:Lunarparallax 22 3 1988.png|thumb|right|250px|Example of lunar parallax: Occultation of Pleiades by the Moon. Credit: .]]
''Lunar parallax'' (often short for ''lunar horizontal parallax'' or ''lunar equatorial horizontal parallax''), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body,  has by far the largest maximum parallax of any celestial body, it can exceed 1 degree.<ref name=aa1981 />

The diagram (above) for stellar parallax can illustrate lunar parallax as well, if the diagram is taken to be scaled right down and slightly modified.  Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and of a circle around the Earth's surface.  Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth:-  one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram);  and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an   Earth-surface position corresponding roughly to one of the blue dots on the modified diagram).

The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth<ref>Astronomical Almanac, e.g. for 1981: see Glossary; for formulae see Explanatory Supplement to the Astronomical Almanac, 1992, p.400</ref> -- equal to angle p in the diagram when scaled-down and modified as mentioned above.

The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth.  The Earth-Moon linear distance varies continuously as the Moon follows its [[w:orbit of the moon|perturbed and approximately elliptical orbit]] around the Earth.  The range of the variation in linear distance is from about 56 to 63.7 earth-radii, corresponding to horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'.<ref name=aa1981>Astronomical Almanac e.g. for 1981, section D</ref> The [[w:Astronomical Almanac|Astronomical Almanac]] and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of [[w:lunar theory|lunar theory]].

Parallax can also be used to determine the distance to the [[Moon]].

One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000&nbsp;km. This procedure was first used by [[w:Aristarchus of Samos|Aristarchus of Samos]]<ref name=Gutzwiller>{{ cite journal
| doi=10.1103/RevModPhys.70.589
| title=Moon-Earth-Sun: The oldest three-body problem
| year=1998
| author=Gutzwiller, Martin C.
| journal=Reviews of Modern Physics
| volume=70
| issue=2
| pages=589
| ref=harv
| bibcode=1998RvMP...70..589G }}</ref> and [[w:Hipparchus|Hipparchus]], and later found its way into the work of [[w:Ptolemy|Ptolemy]]. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed.

Another method is to take two pictures of the Moon at exactly the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated:
:<math>\mathrm{distance}_{\textrm{moon}} = \frac {\mathrm{distance}_{\mathrm{observerbase}}} {\tan (\mathrm{angle})}</math>

==Calculuses==
{{main|Calculuses|Calculus}}
'''Calculus''' uses methods originally based on the summation of infinitesimal differences.

It includes the examination of changes in an expression by smaller and smaller differences.

==Derivatives==
{{main|Effects/Derivatives|Derivatives}}
'''Def.''' a result of an operation of deducing one function from another according to some fixed law is called a '''derivative'''.

Let
: <math>y = f(x)</math>
be a function where values of <math>x</math> may be any real number and values resulting in <math>y</math> are also any real number.

: <math>\Delta x</math> is a small finite difference in <math>x</math> which when put into the function <math>f(x)</math> produces a <math>\Delta y</math>.

These small differences can be manipulated with the operations of arithmetic: addition (<math>+</math>), subtraction (<math>-</math>), multiplication (<math>*</math>), and division (<math>/</math>).

: <math>\Delta y = f(x + \Delta x) - f(x)</math>

Dividing <math>\Delta y</math> by <math>\Delta x</math> and taking the limit as <math>\Delta x</math> → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

: <math>f(x) = x^2</math>

: <math>f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + \Delta x^2</math>

: <math>\Delta y/\Delta x = (x^2 + 2x\Delta x + \Delta x^2 - x^2)/\Delta x</math>

: <math>\Delta y/\Delta x = 2x + \Delta x</math>

as <math>\Delta x</math> and<math>\Delta y</math> go towards zero,

: <math>dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.</math>

This ratio is called the derivative.

==Partial derivatives==
{{main|Effects/Derivatives/Partials|Partial derivatives}}
Let

: <math>y = f(x,z)</math>

then

: <math>\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z</math>

: <math>\partial y/ \partial x = \partial f(x,z)</math>

where z is held constant and

: <math>\partial y / \partial z = \partial f(x,z)</math>

where x is held constant.

==Gradients==
{{main|Spaces/Gradients|Gradients}}
'''Notation''': let the symbol <math>\nabla</math> be the gradient, i.e., derivatives for multivariable functions.

: <math>\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.</math>

==Area under a curve==

Consider the curve in the graph in the section about [[Astronomy/Mathematics#Areas|areas]]. The x-direction is left and right, the y-direction is vertical.

For

: <math>\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x</math>

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

: <math>\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.</math>

Any particular individual rectangle for a sum of rectangular areas is

: <math>f(x_i + \Delta x_i) * \Delta x_i.</math>

The approximate area under the curve is the sum <math>\sum</math> of all the individual (i) areas from i = 0 to as many as the area needed (n):

: <math>\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.</math>

==Integrals==

'''Def.''' a number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed is called an '''integral'''.

'''Notation''': let the symbol <math>\int </math> represent the '''integral'''.

: <math>limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.</math>

This can be within a finite interval [a,b]

: <math>\int_a^b f(x) \; dx</math>

when i = 0 the integral is evaluated at <math>a</math> and i = n the integral is evaluated at <math>b</math>. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

==Theoretical calculus==

'''Def.''' a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a '''calculus'''.

'''Calculus''' focuses on [[w:limit (mathematics)|limits]], [[w:function (mathematics)|functions]], [[w:derivative|derivative]]s, [[w:integral|integral]]s, and [[w:Series (mathematics)|infinite series]].

Although ''calculus'' (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero).

==Line integrals==

'''Def.''' an integral the domain of whose integrand is a curve is called a '''line integral'''.

"The pulsar dispersion measures [(DM)] provide directly the value of

:<math>DM =  \int_0^\infty n_e\, ds</math>

along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure"<ref name=Reynolds>{{ cite journal
|author=R. J. Reynolds
|title=Line Integrals of n<sub>e</sub> and <math>n_e^2</math> at High Galactic Latitude
|journal=The Astrophysical Journal
|month=May 1,
|year=1991
|volume=372
|issue=05
|pages=L17-20
|url=http://adsabs.harvard.edu/full/1991ApJ...372L..17R
|arxiv=
|bibcode=1991ApJ...372L..17R
|doi=10.1086/186013
|pmid=
|accessdate=2013-12-17 }}</ref>

: <math>EM = \int_0^\infty n_e^2 ds.</math>

==Vectors==
{{main|Vectors}}
[[Image:3D Vector.svg|100px|thumb|right]] For standard basis, or unit, vectors ('''i''', '''j''', '''k''') there may be vector components of '''a''' ('''a'''<sub>x</sub>, '''a'''<sub>y</sub>, '''a'''<sub>z</sub>).

'''Def.''' a directed quantity, one with both magnitude and direction; the signed difference between two points is called a '''vector'''.

"An observed time series consists of ''N'' data values x(t<sub>α</sub>) taken at a set of ''N'' discrete times {t<sub>α</sub>}. Hence it defines an ''N''-dimensional ''contravariant vector'' in ''sampling space'', by taking as the α<sup>th</sup> component of the vector, the value of the data at time t<sub>α</sub>, i.e.,
:<math>x^\alpha = [x({t_1}),x({t_2}),...,x({t_N})].</math>
This representation is the ''canonical basis'' for sampling space."<ref name=Foster/>
{{clear}}

==Tensors==
{{main|Tensors}}
'''Def.''' a mathematical object consisting of a set of components with n indices each of which range from 1 to m where n is the rank and m is the dimension is called a '''tensor'''.

"An impressive array of time series analysis methods are equivalent to treating the data as a vector in function space, then projecting the data vector onto a subspace of low dimension. A geometric approach isolates and exposes many of the important features of time series techniques, directly adapts to irregular time spacing, and easily accommodates variable statistical weights. Tensor notation provides an ideal formalism for these techniques. It is quite convenient for distinguishing a variety of different vector spaces, and is the most compact notation for all the sums which arise in the analysis."<ref name=Foster>{{ cite journal
|author=Grant Foster
|title=Time Series Analysis by Projection. II. Tensor Methods for Time Series Analysis
|journal=The Astronomical Journal
|month=January
|year=1996
|volume=111
|issue=1
|pages=555-65
|url=http://adsabs.harvard.edu/full/1996AJ....111..555F
|arxiv=
|bibcode=1996AJ....111..555F
|doi=
|pmid=
|accessdate=2013-12-16 }}</ref>

"[T]he generally invariant line element

: <math> d s^2 = g_{\mu \nu} dx^{\mu}dx^{\nu} </math>

[contains] the spacetime metric tensor <math>g_{\mu \nu} (x^{\rho}), \mu, \nu, \rho = 0, 1, 2, 3,</math> [which] plays a dual role: on the one hand it determines the spacetime geometry, on the other it represents the (ten components of the) gravitational potential, and is thus a dynamical variable."<ref name=Bicak>{{ cite journal
|author=Jiří Bičák
|title=Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics, In: ''Einstein’s Field Equations and Their Physical Implications''
|journal=Lecture Notes in Physics
|publisher=Springer Berlin Heidelberg
|location=Berlin
|month=
|year=2000
|editor=
|volume=540
|issue=
|pages=1-126
|url=http://arxiv.org/pdf/gr-qc/0004016
|arxiv=
|bibcode=
|doi=10.1007/3-540-46580-4_1
|pmid=
|isbn=978-3-540-67073-5
|accessdate=2013-07-04 }}</ref>

==Electronic computers==
{{main|Electronic computers}}
'''Def.''' a programmable electronic device that performs mathematical calculations and logical operations, especially one that can process, store and retrieve large amounts of data very quickly; now especially, a small one for personal or home use employed for manipulating text or graphics, accessing the Internet, or playing games or media is called a '''computer'''.

A '''computer''' is a general purpose device that can be [[w:Computer program|programmed]] to carry out a finite set of arithmetic or logical operations. Since a sequence of operations can be readily changed, the computer can solve more than one kind of problem.

==Programmings==
{{main|Programmings}}
A '''computer program''' (also '''[[w:computer software|software]]''', or just a '''program''') is a sequence of [[w:instruction (computer science)|instructions]] written to perform a specified task with a computer.<ref name="pis-ch4-p132">{{ cite book
  | last = Stair
  | first = Ralph M.
  |display-authors=etal
  | title = Principles of Information Systems, Sixth Edition
  | publisher = Thomson Learning, Inc.
  | date = 2003
  | pages = 132
  | isbn = 0-619-06489-7 }}</ref> A computer requires programs to function, typically [[w:execution (computing)|executing]] the program's instructions in a [[w:central processing unit|central processor]].<ref name="osc-ch3-p58">{{ cite book
  | last = Silberschatz
  | first = Abraham
  | title = Operating System Concepts, Fourth Edition
  | publisher = Addison-Wesley
  | date = 1994
  | pages = 58
  | isbn = 0-201-50480-4 }}</ref>

'''Computer programming''' (often shortened to '''programming''' or '''coding''') is the process of [[w:Software design|designing]], writing, [[w:Software testing|testing]], [[w:debugging|debugging]], and maintaining the [[w:source code|source code]] of [[w:computer program|computer program]]s.

==Probabilities==
{{main|Probabilities}}
'''Def.''' a number, between 0 and 1, expressing the precise likelihood of an event happening is called a '''probability'''.

'''Probability''' is a measure of the expectation that an event will occur or a statement is true. Probabilities are given a value between 0 (will not occur) and 1 (will occur).<ref name=Feller>{{ cite book
|author=William Feller
|date=1968
|title=An Introduction to Probability Theory and its Applications
|volume=1
|ISBN=0-471-25708-7
}}</ref>  The higher the probability of an event, the more certain we are that the event will occur.

==Statistics==
{{main|Statistics}}
'''Def.''' a  mathematical science concerned with data collection, presentation, analysis, and interpretation is called '''statistics'''.

'''[[Statistics]]''' is the study of the collection, organization, analysis, interpretation, and presentation of [[w:data|data]].<ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}}</ref><ref>[http://www.thefreedictionary.com/dict.asp?Word=statistics The Free Online Dictionary]</ref> It deals with all aspects of this, including the planning of data collection in terms of the design of [[w:statistical survey|survey]]s and [[w:experimental design|experiments]].<ref name=Dodge/>

"Statistics of projections are derived under a number of different null hypotheses."<ref name=Foster/>

==Hypotheses==
{{main|Hypotheses}}
# Each mathematical approach requires a proof of concept.

==See also==
{{div col|colwidth=12em}}
* [[w:Cosmic distance ladder|Cosmic distance ladder]]
* [[Portal:Euclidean geometry|Euclidean geometry]]
* [[Topic:Mathematical physics|Mathematical physics]]
* [[Portal:Mathematics|Mathematics]]
* [[Radiation astronomy/Courses/Principles|Principles of Radiation Astronomy]]
* [[Probability]]
* [[Radiation astronomy/Mathematics|Radiation mathematics]]
* [[Statistics]]
* [[Astronomy/Theory|Theoretical astronomy]]
* [[Radiation astronomy/Theory|Theoretical radiation astronomy]]
* [[Topic:Trigonometry|Trigonometry]]
{{Div col end}}

==References==
{{reflist|2}}

==Further reading==
* {{ cite book
|author=William Marshall Smart
|author2=Robin Michael Green
|title=Textbook on Spherical Astronomy, Sixth Edition
|publisher=University of Cambridge
|location=Cambridge
|date=July 7, 1977
|editor=
|pages=431
|url=http://books.google.com/books?id=W0f2vc2EePUC&lr=&source=gbs_navlinks_s
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=0 521 21516 1
|accessdate=2012-05-18 }}
* {{ cite journal 
| title=Large-scale expanding superstructures in galaxies 
| author=Tenorio-Tagle G
|author2=Bodenheimer P
| journal=Annual Review of Astronomy and Astrophysics 
| date=1988 
|volume=26 
|issue= 
|pages=145–97 
|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1988ARA%26A..26..145T 
}}

==External links==
* [http://www.bing.com/search?q=&go=&qs=n&sk=&sc=8-15&qb=1&FORM=AXRE Bing Advanced search]
* [http://books.google.com/ Google Books]
* [http://scholar.google.com/advanced_scholar_search?hl=en&lr= Google scholar Advanced Scholar Search]
* [http://www.iau.org/ International Astronomical Union]
* [http://www.jstor.org/ JSTOR]
* [http://www.lycos.com/ Lycos search]
* [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED]
* [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center.]
* [http://www.questia.com/ Questia - The Online Library of Books and Journals]
* [http://online.sagepub.com/ SAGE journals online]
* [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System]
* [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search]
* [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer]
* [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database]
* [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA]
* [http://www.springerlink.com/ SpringerLink]
* [http://www.tandfonline.com/ Taylor & Francis Online]
* [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter]
* [http://onlinelibrary.wiley.com/advanced/search Wiley Online Library Advanced Search]
* [http://search.yahoo.com/web/advanced Yahoo Advanced Web Search]

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