Mathematical astronomy

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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.

Educational level: this is a secondary education resource.

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

Educational level: this is a tertiary (university) resource.

Educational level: this is a research resource.

Resource type: this resource is an article.

Resource type: this resource contains a lecture or lecture notes.

Subject classification: this is an astronomy resource.

Subject classification: this is a mathematics resource .

Notation

Notation: let the symbol Def. indicate that a definition is following.

Notation: let the symbol $R_\oplus$ indicate the Earth's radius.

Notation: let the symbol $R_{\odot}$ indicate the solar radius.

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^-2 or x².

In the Notations at the top of this section, Index is replaced by symbols for the Sun (ʘ), Earth ($R_\oplus$), or can be for Jupiter (J) such as $R_J$.

A common Oversymbol is one for the average $\overline{Variable}$.

Operation may be replaced by a function, for example.

All notational locations could look something like

bx	$-$	x = n
a	$\sum$	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.

Universals

For definitions, their meanings and intents, there is the learning resource theory of definition.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

Def. "[a]n abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts", from Wiktionary mathematics, is called mathematics.

A nomy (Latin nomia) is a "system of laws governing or [the] sum of knowledge regarding a (specified) field." Nomology is the "science of physical and logical laws."

While in astronomy most entities have names, demonstrating that one or more numbers, shapes, structures, or changes are associated with an entity is evidence of proof of concept that mathematics is applicable to astronomy.

Astronomical units

Def. "1 day (d)" is called the astronomical unit of time.

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).

Def. "the distance at which one Astronomical Unit subtends an angle of one arcsecond" is called the parsec (pc).

Def. "365.25 days" is called a Julian year.

Def. "36,525 days" is called a Julian century.

Dimensional analysis

Def. "[a] single aspect of a given thing", per Wiktionary dimension, 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 10⁸ km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 10⁸ is a number.

Def. "[t]he study of the dimensions of ... quantities; used to obtain information about large complex systems, and as a means of checking ... equations", after Wiktionary dimensional analysis, is called dimensional analysis.

Algebra

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

Def. "[t]he mathematics of numbers (integers, rational numbers, real numbers, or complex numbers) under the operations of addition, subtraction, multiplication, and division", per Wiktionary arithmetic, is called an arithmetic.

Notation: let the symbol * designate an as yet unspecified operation.

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

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).

Def. "[a] system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols", from Wiktionary algebra, is called an algebra.

The manipulations of these symbols are performed using operations.

Def. "a procedure for generating a value from one or more other values (the operands; the value for any particular [operand] is unique)", from Wiktionary operation, is called an operation.

The results are recorded using statements of relation.

Cosmic distance ladder

"The 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 kpc equals 1000 parsecs) as follows:

$\begin{smallmatrix}5 \cdot \log_{10} \frac{D}{\mathrm{kpc}}\ =\ m\ -\ M\ -\ 10,\end{smallmatrix}$

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

Logical law

The diagram illustrates Kepler's three laws using two planetary orbits. Credit: Hankwang.

"Kepler's laws:

1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

The diagram at the right illustrates Kepler's three laws of planetary orbits: (1) The orbits are ellipses, with focal points ƒ₁ and ƒ₂ for the first planet and ƒ₁ and ƒ₃ for the second planet. The Sun is placed in focal point ƒ₁. (2) The two shaded sectors A₁ and A₂ have the same surface area and the time for planet 1 to cover segment A₁ is equal to the time to cover segment A₂. (3) The total orbit times for planet 1 and planet 2 have a ratio a₁^3/2 : a₂^3/2.

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.

Inverse image

Def. “[t]he set of points that map to a given point (or set of points) under a specified function”, from Wiktionary inverse image, is called an inverse image.

“Under the function given by $f(x)=x^2$, the inverse image of 4 is $\{-2,2\}$, as is the inverse image of $\{4\}$", after Wiktionary inverse image.

Conic sections

Conics are of three types: parabolas , ellipses, including circles, and hyperbolas. Credit: .

Conic parameters in the case of an ellipse

"In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane." from the Wikipedia entry about the conic section.

"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.)" per the Wikipedia entry about the conic section.

conic section	equation	eccentricity (e)	linear eccentricity (c)	semi-latus rectum (ℓ)	focal parameter (p)
circle	$x^2+y^2=a^2 \,$	$0 \,$	$0 \,$	$a \,$	$\infty$
ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	$\sqrt{1-\frac{b^2}{a^2}}$	$\sqrt{a^2-b^2}$	$\frac{b^2}{a}$	$\frac{b^2}{\sqrt{a^2-b^2}}$
parabola	$y^2=4ax \,$	$1 \,$	$a \,$	$2a \,$	$2a \,$
hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$\sqrt{1+\frac{b^2}{a^2}}$	$\sqrt{a^2+b^2}$	$\frac{b^2}{a}$	$\frac{b^2}{\sqrt{a^2+b^2}}$

Orbits

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: Stamcose.

"Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions.

In the geocentric model of the solar system, the 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 deferent and epicycles 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.", from the Wikipedia article Orbit.

In 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 solar system with the 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—namely as parallactic displacements resulting from the Earth's motion around the Sun—an important consideration in Johannes Kepler's conviction that the theory was substantially correct." "[Kepler] knew that the tables constructed from the heliocentric theory were more accurate than those of Ptolemy" 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.

Two bodies orbiting around a common barycenter (red cross) with circular orbits. Credit: Zhatt.

Two bodies orbiting around a common barycenter (red cross) with elliptic orbits. Credit: Zhatt.

ISEE-3 is inserted into a "halo" orbit on June 10, 1982. Credit: NASA.

Antapex

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

"[T]he local standard of rest or LSR follows the mean motion of material in the Milky Way in the neighborhood of the Sun. The path of this material is not precisely circular. The Sun follows the solar circle (eccentricity e < 0.1 ) at a speed of about 220 km/s in a clockwise direction when viewed from the galactic north pole at a radius of ≈ 8 kpc about the center of the galaxy near Sgr A*, and has only a slight motion, towards the solar apex, relative to the LSR. [The Sun's peculiar motion relative to the LSR is 13.4 km/s.] The LSR velocity is anywhere from 202–241 km/s."

Eccentricity

"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 10⁵ yr". "Smaller amplitude variations occur with about a 10⁵ yr period."

Inclination

The diagram describes the parameters associated with orbital inclination (i). Credit: Lasunncty.

"The orbital inclination [i] [of Mercury] varies between 5° and 10° with a 10⁶ yr period with smaller amplitude variations with a period of about 10⁵ yr."

Obliquity

"[A]xial tilt (also called obliquity) is the angle between an object's rotational axis, and a line perpendicular to its orbital plane. ... The planet Venus has an axial tilt of 177.3° because it is rotating in retrograde direction, opposite to other planets like Earth. ... The planet 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 Sun. Hence the axial tilt of Uranus is 97°."

The obliquity of the Earth's axis has a period of about 41,000 years.

Precession

The equinoxes of Earth precess with a period of about 21,000 years.

Spherical geometry

Def. "[t]he non-Euclidean geometry on the surface of a sphere", is called spherical geometry.

Horizontal coordinate system

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 celestial coordinate system that uses the observer's local horizon as the fundamental plane. This coordinate system divides the sky into the upper hemisphere where objects are visible, and the lower hemisphere where objects cannot be seen since the earth is in the way. The great circle separating hemispheres [is] called [the] celestial horizon or rational horizon. The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir. ”

“The horizontal coordinates are:

• Altitude (Alt), sometimes referred to as 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.
• 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 complement of altitude (i.e. 90°-altitude).”

Fixed point in the sky

By choosing an equal day/night position among the fixed objects in the night sky, the observer can measure equatorial coordinates: declination (Dec) and right ascension (RA). Credit: .

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 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 equatorial coordinates: declination (Dec) and 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 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.

Trigonometry

Def. "the relationships between the sides and the angles of triangles and the calculations based on them", after Wiktionary trigonometry, is called trigonometry.

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

$\beta_{app} = { \beta_{jet} \sin \phi \over 1 - \beta_{jet} \cos \phi },$

where β_jetc is the jet velocity, and the jet makes an angle Φ to the line of sight."

Radius of the Earth

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

Distance computation

Stellar parallax motion.

"Distance measurement by parallax is a special case of the principle of 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 arcsecond, 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 derivation below), the distance to an object (measured in parsecs) is the reciprocal of the parallax (measured in arcseconds): $d (\mathrm{pc}) = 1 / p (\mathrm{arcsec}).$ For example, the distance to Proxima Centauri is 1/0.7687=1.3009 parsecs (4.243 ly)." per parallax.

Distance to the Moon

Any distance to the Moon is often initially calculated as a multiple of the Earth radius $R_\oplus$.

Parallax

"Parallax is a displacement or difference in the 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.""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.""

"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 stars 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 terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.""

Lunar parallax

Diagram of daily lunar parallax. 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.

Example of lunar parallax: Occultation of Pleiades by the Moon. Credit: .

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 -- 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 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'. The 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 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 km. This procedure was first used by Aristarchus of Samos and Hipparchus, and later found its way into the work of 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:

$\mathrm{distance}_{\textrm{moon}} = \frac {\mathrm{distance}_{\mathrm{observerbase}}} {\tan (\mathrm{angle})}$"

See also

• Astronomy
• Cosmic distance ladder
• Euclidean geometry
• Mathematical physics
• Mathematics
• Principles of Radiation Astronomy
• Probability
• Statistics
• Theoretical astronomy
• Theoretical radiation astronomy
• Trigonometry

Further reading

• William Marshall Smart, Robin Michael Green (July 7, 1977). Textbook on Spherical Astronomy, Sixth Edition. Cambridge: University of Cambridge. pp. 431. ISBN 0 521 21516 1. http://books.google.com/books?id=W0f2vc2EePUC&dq=calculus+astronomy&lr=&source=gbs_navlinks_s. Retrieved 2012-05-18. 
• Tenorio-Tagle G, Bodenheimer P (1988). "Large-scale expanding superstructures in galaxies". Annual Review of Astronomy and Astrophysics 26: 145–97. 

External links

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