Radiation mathematics

 

This animation depicts the collision between our Milky Way galaxy and the Andromeda galaxy. Credit: Visualization Credit: NASA; ESA; and F. Summers, STScI; Simulation Credit: NASA; ESA; G. Besla, Columbia University; and R. van der Marel, STScI.

Completion status: About halfway there. You may help to clarify and expand it.

Most of the mathematics needed to understand the information acquired through astronomical radiation observation comes from physics. But, there are special needs for situations that intertwine mathematics with phenomena that may not yet have sufficient physics to explain the observations. Both uses constitute radiation mathematics, or astronomical radiation mathematics, or a portion of mathematical radiation astronomy.

Educational level: this is a secondary education resource.

Astronomical radiation mathematics is the laboratory mathematics such as simulations that are generated to try to understand the observations of radiation astronomy.

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

The mathematics needed to understand radiation astronomy starts with arithmetic and often needs various topics in calculus and differential equations to produce likely models.

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 .

Subject classification: this is a statistics resource .

Subject classification: this is a technology resource .

Notation

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

Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.

Notation: let the symbol $L_{\odot}$ indicate the solar visible luminosity.

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

To help with 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.

The proof-of-concept structure consists of

1. background,
2. procedures,
3. findings, and
4. interpretation.

The findings demonstrate a statistically systematic change from the status quo or the control group.

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.

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.

Numbers

Notation: let PeV denote 10¹⁵ eV.

Notation: let EeV denote 10¹⁸ eV.

“Outside the nucleus, free neutrons are unstable and have a mean lifetime of 885.7±0.8 s (about 14 minutes, 46 seconds); therefore the half-life for this process (which differs from the mean lifetime by a factor of ln(2) = 0.693) is 613.9±0.8 s (about 10 minutes, 11 seconds)."

202 stars complete the magnitude range 8.90 < V < 16.30.

Astronomical units

Def. "the distance traveled by light in one Julian year in a vacuum" is called the light-year (ly).

Arithmetic

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.

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.

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 binary star. The operation of addition here usually requires an explanation (a theory).

Arithmetic can apply to tables of numbers by pairing the numbers together. A concept of relative intensity might cover the range from zero to 100, where the maximum observed intensity in dimensional units is eliminated by dividing each of the observed intensities by the maximum.

55 photons, 22 photons, 11 photons divided by 55 photons yields 1, 0.4, 0.2, or when times 100: 100, 40, and 20. If each of these intensities (photons in this case) occurred over different types of radiation (X-ray, Gamma ray, and visual), then binary pairs can be created:

1. 100 X-rays
2. 40 gamma rays
3. 20 visual rays, for example.

As each band may have an average wavelength, the pairs can become

1. 100 5 nm
2. 40 0.5 pm
3. 20 500 nm,

which can be ordered by wavelength and graphed to show a spectrum.

Arithmetic can also be performed at various notational locations.

^{100 kg}n^{0 coul} → ^{99 kg}p^{+1 coul} + ^{1 kg}e^{-1 coul},

100 kg → 99 kg + 1 kg, and

0 coul of net charge → +1 coul + -1 coul of separated charges.

4(X-rays of 1 nm) + 5(X-rays of 1 nm) = (4 + 5)(X-rays of 1 nm) = 9(X-rays of 1 nm),

2x * 1y → (2x,1y), a binary pair, or

_nΩ + _{1 to ∞}Ω → _n=1-∞Ω.

Ψ_i=1 + Ψ_i=2 = Ψ_i=1-2.

But, the exponent can require a different operator of arithmetic.

e² + e³ ≠ e⁵. Yet

e² x e³ = e^{(2 + 3)} = e⁵.

Here, the context determines the operation.

Free neutron decay

"Free neutrons decay by emission of an electron and an electron antineutrino to become a proton, a process known as beta decay:”

n0
=> p+
+ e−
+ ν
e

For the above relation

Notational locations
Weight	Oversymbol	Exponent
Coefficient	Variable	Operation
Number	Range	Index

Starting with the left symbol, Weight is 1 (not mentioned), Oversymbol is not used, Exponent is replaced by Charge, the Coefficient is 1 (not mentioned), the Variable is a letter designation for the subatomic particle of interest (n for neutron), the Operation is actually a relation decays to (=>), Number is the atomic number Z = 0 for a neutron (not mentioned), the Range is not applicable, and no Index is being used. The neutron's decay products are a proton (p), electron (e), and a neutrino (ν), where Index is used to indicate that the neutrino is an electron neutrino and Oversymbol indicates it is actually an antineutrino. The Operation (+) is not mathematical addition, but indicates another decay product.

Neutron ejection

“Gamma radiation with an energy exceeding the neutron binding energy of a nucleus can eject a neutron. Two examples and their decay products:”

⁹Be + >1.7 Mev photon → 1 neutron + 2 ⁴He

²H (deuterium) + >2.26 MeV photon → 1 neutron + ¹H

Algebra

Fundamentally, algebra uses letters to represent as yet unspecified numbers. The numbers may be integers, rational numbers, irrational numbers, or any real number or complex number. As an 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.

For radiation, there are spatial, temporal, energy, or wavelength choices, among others. Meteors penetrating some portion of a planet's atmosphere can be said to have energy, a spatial distribution, a temporal one, or be in a relationship to something else.

Binary pairs of the radiation, perhaps expressed as meteors/km² at progressive time intervals: (meteors/km²,years). The km² is a surface area through which some number of meteors are observed. One may state: each year over Madrid 200 meteors are observed, that would be an arithmetic fact of observation. Theoretically, this could be M km² per year, where M indicates the air space over Madrid divided into squares so many kilometers on a side.

Your theory would relate M to your explanation for the meteors. Years may be expressed as Julian years, years AD, b2k, or another dimension.

Degrees of freedom

"For other distributions, two-thirds of the average energy is often referred to as the temperature, since for a Maxwell-Boltzmann distribution with three degrees of freedom, $\langle E \rangle = (3/2) \langle k_BT \rangle$."

Metallicity

Blue main sequence stars that are metal poor ([Fe/H] ≤ -1.0) are most likely not analogous to blue stragglers.

"The metallicity of the Sun is approximately 1.8 percent by mass. For other stars, the metallicity is often expressed as "[Fe/H]", which represents the logarithm of the ratio of a star's iron abundance compared to that of the Sun (iron is not the most abundant heavy element, but it is among the easiest to measure with spectral data in the visible spectrum). The formula for the logarithm is expressed thus:

$[\mathrm{Fe}/\mathrm{H}] = \log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{star}} - \log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{sun}}$

where $N_{\mathrm{Fe}}$ and $N_{\mathrm{H}}$ are the number of iron and hydrogen atoms per unit of volume respectively. The unit often used for metallicity is the "dex" which is a (now-deprecated) contraction of decimal exponent. By this formulation, stars with a higher metallicity than the Sun have a positive logarithmic value, while those with a lower metallicity than the Sun have a negative value. The logarithm is based on powers of ten; stars with a value of +1 have ten times the metallicity of the Sun (10¹). Conversely, those with a value of -1 have one tenth (10 ⁻¹), while those with -2 have a hundredth (10⁻²), and so on. Young Population I stars have significantly higher iron-to-hydrogen ratios than older Population II stars. Primordial Population III stars are estimated to have a metallicity of less than −6.0, that is, less than a millionth of the abundance of iron which is found in the Sun."

Hydrogen spectra

"The energy differences between levels in the Bohr model, and hence the wavelengths of emitted/absorbed photons, is given by the Rydberg formula:

${1 \over \lambda} = R \left( {1 \over (n^\prime)^2} - {1 \over n^2} \right) \qquad \left( R = 1.097373 \times 10^7 \ \mathrm{m}^{-1} \right)$

where n is the initial energy level, n′ is the final energy level, and R is the Rydberg constant. Meaningful values are returned only when n is greater than n′ and the limit of one over infinity is taken to be zero."

Chemical composition

"Olivines are described by Mg_2yFe_2-2ySiO₄, with y in[0, 1]." Substituting values for y from 0 to 1 produce ideal compositions from forsterite Mg₂SiO₄ to fayalite Fe₂SiO₄. "Amorphous olivine with y = 0.5 and crystalline olivine with y = 0.95 were taken into account for the olivine component." as best fits to observed data. "In the green, the polarization of the pure silicate composition qualitatively appears a better fit to the shape of the observed polarization curves". The silicates used to model the cometary coma dust are olivene (Mg-rich is green) and the pyroxene, enstatite.

Exponential decrease

"When a gamma ray passes through matter, the probability for absorption is proportional to the thickness of the layer, the density of the material, and the absorption cross section of the material. The total absorption shows an exponential decrease of intensity with distance from the incident surface:

$I(x) = I_0 \cdot e ^{-\mu x}$

where μ = nσ is the absorption coefficient, measured in cm⁻¹, n the number of atoms per cm³ in the material, σ the absorption cross section in cm² and x the thickness of material in cm."

"The time evolution of the number of emitted scintillation photons N in a single scintillation event can often be described by the linear superposition of one or two exponential decays. For two decays, we have the form:

$N = A\exp\left(-\frac{t}{{\tau}_f}\right) + B\exp\left(-\frac{t}{{\tau}_s}\right)$

where τ_f and τ_s are the fast (or prompt) and the slow (or delayed) decay constants. Many scintillators are characterized by 2 time components: one fast (or prompt), the other slow (or delayed). While the fast component usually dominates, the relative amplitude A and B of the two components depend on the scintillating material. Both of these components can also be a function the energy loss dE/dx."

"In cases where this energy loss dependence is strong, the overall decay time constant varies with the type of incident particle. Such scintillators enable pulse shape discrimination, i.e., particle identification based on the decay characteristics of the PMT electric pulse. For instance, when BaF₂ is used, gamma rays typically excite the fast component, while alpha particles excite the slow component: it is thus possible to identify them based on the decay time of the PMT signal."

"For cold dust clouds in the interstellar medium β is approximately two. Therefore Q becomes,

$Q_{\nu}=1-e^{-\tau_{\nu}}=1-e^{-\tau_0 (\nu / \nu_{0})^{\beta}}$. ($\tau_0=1$, $\nu_0$ is the frequency where $\tau_0=1$)".

Lorentz factor

"The Lorentz factor is defined as:

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau}$

where:

• v is the relative velocity between inertial reference frames,
• β is the ratio of v to the speed of light c.
• τ is the proper time for an observer (measuring time intervals in the observer's own frame),
• c is the speed of light."

Dose equivalent

From the Wikipedia article on the sievert, “The equivalent dose to a tissue is found by multiplying the absorbed dose, in gray, by a weighting factor (W_R). The relation between absorbed dose D and equivalent dose H is thus:

$H = W_R \cdot D$.

The weighting factor (sometimes referred to as a quality factor) is determined by the radiation type and energy range.

$H_T = \sum_R W_R \cdot D_{T,R}\ ,$

where

H_T is the equivalent dose absorbed by tissue T

D_T,R is the absorbed dose in tissue T by radiation type R

W_R is the weighting factor defined by the following table

Radiation type and energy		WR
electrons, muons, photons (all energies)		1
protons and charged pions		2
alpha particles, fission fragments, heavy ions		20
neutrons (function of linear energy transfer L in keV/μm)	L < 10	1
	10 ≤ L ≤ 100	0.32·L − 2.2
	L > 100	300 / sqrt(L)

Thus for example, an absorbed dose of 1 Gy by alpha particles will lead to an equivalent dose of 20 Sv. The maximum weight of 30 is obtained for neutrons with L = 100 keV/μm.”

Effective dose

“The effective dose of radiation (E), absorbed by a person is obtained by averaging over all irradiated tissues with weighting factors adding up to 1:

$E = \sum_T W_T \cdot H_T = \sum_T W_T \sum_R W_R \cdot D_{T,R}$.

Electronic computers

"A computer is a general purpose device that can be 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."

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). The higher the probability of an event, the more certain we are that the event will occur."

Def. "[a] family of continuous probability distributions such that the probability density function is the Gaussian function

1. $\varphi_{\mu,\sigma^2}(x) = \frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x- \mu)^2}{2\sigma^2}} = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right),\quad x\in\mathbb{R}$" is called a normal distribution.

Programming

"A computer program (also software, or just a program) is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor."

"Computer programming (often shortened to programming or coding) is the process of designing, writing, testing, debugging, and maintaining the source code of computer programs."

Statistics

"Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments."

Geometry

The universe as often perceived may be described spatially, sometimes with plane geometry, other occasions with spherical geometry.

Trigonometry

"Trigonometry ... studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves."

Calculus

"Calculus [focuses] on limits, functions, derivatives, integrals, and infinite series."

Solar limb brightness

Def. "the radial distance q from the Sun's center such that the following finite Fourier transform is zero:

$F(G; q, a) = \int_{-1/2}^{+1/2} G(q + a \sin \pi s) \cos(2\pi s) ds,$

where s is a dummy variable, G is the observed solar intensity as a function of the radius, and the parameter a determines the extent of the solar limb used" is called the solar edge.

"When F(G; q, a) = 0, the a dependence of q can be used to choose different points as the edge."

Radiation source

For some plasma sources, "an exponential spectrum corresponding to a thermal bremsstrahlung source [may fit]":

N(E)dE = E₀^-1 * exp^-E/kTdE.

dN/dE = (E₀/E) * exp^-E/kT, where a least squares fit to the radiated detection data yields a kT.

Another equation used to study astronomical events is the power law:

$f(x) = ax^k$,.

In terms of radiation detected, for example, f(x) = photons (cm²-sec-keV)^-1 versus keV. As the photon flux decreases with increasing keV, the exponent (k) is negative. Observations of X-rays have sometimes found the spectrum to have an upper portion with k ~ -2.3 and the lower portion being steeper with k ~ -4.7. This suggests a two stage acceleration process.

“[T]he synchrotron functions are defined as follows (for x ≥ 0):

• First synchrotron function

$F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt$

• Second synchrotron function

$G(x) = x K_{\frac{2}{3}}(x)$

where K_j is the modified Bessel function of the second kind. The function F(x) is shown on the right, as the output from a plot in Mathematica.

First synchrotron function, F(x)

In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.”

Peak wavelength for a Planckian radiator

Planck's equation (colored curves) accurately describes black body radiation. Credit: Darth Kule.

"Planck's [equation] describes the amount of [spectral radiance at] a certain wavelength radiated by a black body in thermal equilibrium".

"In terms of ... wavelength (λ), Planck's [equation] is written:[ as]

$B_\lambda(T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$

where B is the spectral radiance, T is the absolute temperature of the black body, k_B is the Boltzmann constant, h is the Planck constant, and c is the speed of light."

This form of the equation contains several constants that are usually not subject to variation with wavelength. These are h, c, and k_B. They may be represented by simple coefficients: c1 = 2h c² and c2 = h c/k_B.

By setting the first partial derivative of Planck's equation in wavelength form equal to zero, iterative calculations may be used to find pairs of (λ,T) that to some significant digits represent the peak wavelength for a given temperature and vice versa.

$\frac{\partial B}{\partial \lambda} = \frac{c1}{\lambda^6}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}[\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} - 5] = 0.$

Or,

$\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} - 5 = 0.$

$\frac{c2}{\lambda T}\frac{1}{ e^{\frac{c2}{\lambda T}} - 1}e^{\frac{c2}{\lambda T}} = 5.$

Use c2 = 1.438833 cm K.

See also

• Astrochemistry
• Astrogeography
• Astrophysics
• Mathematical astronomy
• Theoretical astronomy
• Theoretical radiation astronomy

Further reading

• K. J. Frost and B. R. Dennis (May 1, 1971). "Evidence from Hard X-Rays for Two-Stage Particle Acceleration in a Solar Flare". The Astrophysical Journal 165 (5): 655. doi:10.1086/150932. Bibcode: 1971ApJ...165..655F. Retrieved on 2012-03-01. 

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