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.
Astronomical radiation mathematics is the laboratory mathematics such as simulations that are generated to try to understand the observations of radiation astronomy.
The mathematics needed to understand radiation astronomy starts with arithmetic and often needs various topics in calculus and differential equations to produce likely models.
Notation: let the symbol 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 (⊙), Earth ( ), or can be for Jupiter (J) such as .
A common Oversymbol is one for the average .
Operation may be replaced by a function, for example.
All notational locations could look something like
bx x = n a 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.
An initial use of mathematics in astronomy is counting 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.
Most of the topics in mathematics that are applied to astronomy have been described in mathematical astronomy
Def. an "abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts" is called mathematics.
Theoretical radiation mathematicsEdit
Def. an action or process of throwing or sending out a traveling ray in a line, beam, or stream of small cross section is called radiation.
The term radiation is often used to refer to the ray itself.
Def. the "shooting forth of anything from a point or surface, like the diverging rays of light; as, the radiation of heat" is called radiation.
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 108 km on average from the Sun. Kilometer (km) is a dimension and 1.50 x 108 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.
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.
Def. "the distance traveled by light in one Julian year in a vacuum" is called the light-year (ly).
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.
- 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:
- 100 X-rays
- 40 gamma rays
- 20 visual rays, for example.
As each band may have an average wavelength, the pairs can become
- 100 5 nm
- 40 0.5 pm
- 20 500 nm,
Arithmetic can also be performed at various notational locations.
- 100 kgn0 coul → 99 kgp+1 coul + 1 kge-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.
- Ψi=1 + Ψi=2 = Ψi=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 decayEdit
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.
- 9Be + >1.7 Mev photon → 1 neutron + 2 4He
- 2H (deuterium) + >2.26 MeV photon → 1 neutron + 1H
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 freedomEdit
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, .
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:
where and 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.
The energy differences between levels in the Bohr model, and hence the wavelengths of emitted/absorbed photons, is given by the Rydberg formula:
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.
"Olivines are described by Mg2yFe2-2ySiO4, 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.
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:
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:
where τf and τs 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.
The monochromatic flux density radiated by a greybody at frequency through solid angle is given by where is the Planck function for a blackbody at temperature T and emissivity .
For a uniform medium of optical depth radiative transfer means that the radiation will be reduced by a factor . The optical depth is often approximated by the ratio of the emitting frequency to the frequency where all raised to an exponent β.
For cold dust clouds in the interstellar medium β is approximately two. Therefore Q becomes,
. ( , is the frequency where ).
The Lorentz factor is defined as:
Def. the dose received in one hour at a distance of 1 cm from a point source of 1 mg of radium in a 0.5 mm thick platinum enclosure is called a sievert.
Equivalent dose to a tissue is found by multiplying the absorbed dose, in gray, by a weighting factor (WR). The relation between absorbed dose D and equivalent dose H is thus:
The weighting factor (sometimes referred to as a quality factor) is determined by the radiation type and energy range.
- HT is the equivalent dose absorbed by tissue T
- DT,R is the absorbed dose in tissue T by radiation type R
- WR 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|
(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.
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 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.
- is called a normal distribution.
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.
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.
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.
"The nuclear processes that produce cosmogenic 36Cl in rocks are spallation, neutron capture, and muon capture. The first two processes dominate production on the land surface; muon production in Ca and K becomes more important with increasing depth (Rama and Honda, 1961)."
"The decay of radioactive U and Th also give rise to the production of 36Cl, via neutron capture (Bentley et al., 1986)."
"The production rate of cosmogenic 36Cl in bedrock and regolith exposed at Earth's surface is dependent on its calcium, potassium, and chloride content and can be expressed by the equation
where and are the total production rates (including production due to slow negative muons) of 36Cl due to potassium and calcium, respectively; and are the elemental concentrations of potassium and calcium, respectively; and is the thermal neutron capture rate, which is dependent on the fraction of neutrons stopped by 35Cl as determined by the effective cross sections of 35Cl and all other absorbing elements and their respective abundances and ."
Theoretical radiation astromathematicsEdit
The overall theory of astronomy consists of three fundamental parts:
- the derivation of logical laws,
- the definitions of natural bodies (entities, sources, or objects), and
- the definition of the sky (and associated realms).
Here's a theoretical definition:
Def. the mathematics that describes the logical laws regarding radiation and astronomical entities, sources, and objects is called theoretical radiation astronomy mathematics, or theoretical radiation astromathematics.
"There is no correlation between n (the exponent for the change in particle diameter) and any of the other entities, which shows that n-values cannot be used as an index of maturation of a soil."
where is change in particle diameter, is the constant of surface area rate of change, is the measured particle diameter in the soil, is the initial particle diameter for all samples, and is the exponent.
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:
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.
The synchrotron functions are defined as follows (for x ≥ 0):
- First synchrotron function
- Second synchrotron function
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 radiatorEdit
In terms of wavelength (λ), Planck's equation is written as
This form of the equation contains several constants that are usually not subject to variation with wavelength. These are h, c, and kB. They may be represented by simple coefficients: c1 = 2h c2 and c2 = h c/kB.
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.
Use c2 = 1.438833 cm K.
Coulomb's law states that the electrostatic force experienced by a charge, at position , in the vicinity of another charge, at position , in vacuum is equal to:
where is the electric constant or the permittivity of free space and is the distance between the two charges and the constant is in SI units of C2 m−2 N−1, where C is Coulombs.
ε0 ≈ 8.854 x 10-12 C2 m−2 N−1, or
ε0 ≈ 8.854 x 10-12 C2 10-6 km−2 N−1, then
ε0 ≈ 8.854 x 10-18 C2 km−2 N−1.
1 e.s.u. ≈ 3.34 x 10-10 C.
Each of the constants used are approximates so Fq ≈ 8.02 x 1028 N.
If the velocity noted on approach is the vector component pointed toward the Sun, then the angle between the velocity vector and the magnetic field of the Sun is about 180°.
Approximately, FqB ≈ 4.01 x 1017 N.
The total Lorentz force is predominantly the electrostatic force.
"The prismoidal method provides a good approximation of the dust emission peak for cold sources [...], but probably overestimates the long-wavelength flux for warmer, flatter [spectral energy distribution] SED sources [...], resulting in a bias toward lower Tbol."
"In a binary formed via gravitational fragmentation, we would expect the separation to correspond to the local Jeans length (Jeans 1928):"
"where cs is the local sound speed, and μp = 2.33 and n are the mean molecular weight and mean particle density, respectively. A Jeans length of 4200 AU would require a relatively high density (n ~ 6 x 105 cm−3, assuming cs = 0.2 km s−1). The mean density of the Per-Bolo 102 core, measured within an aperture of 104 AU, is 4 x 105 cm−3, close to the required value."
Alpha decay is characterized by the emission of an alpha particle, a 4He nucleus. The mode of this decay causes the parent nucleus to decrease by two protons and two neutrons. This type of decay follows the relation:
Neutron activation is the process in which neutron radiation induces radioactivity in materials, and occurs when atomic nuclei capture free neutrons, becoming heavier and entering excited states. The excited nucleus often decays immediately by emitting particles such as neutrons, protons, or alpha particles. The neutron capture, even after any intermediate decay, often results in the formation of an unstable activation product. Such radioactive nuclei can exhibit half-lives ranging from small fractions of a second to many years.
Proton emission (also known as proton radioactivity) is a type of radioactive decay in which a proton is ejected from a nucleus. Proton emission can occur from high-lying excited states in a nucleus following a beta decay, in which case the process is known as beta-delayed proton emission, or can occur from the ground state (or a low-lying isomer) of very proton-rich nuclei, in which case the process is very similar to alpha decay.
Beta decay is characterized by the emission of a neutrino and a negatron which is equivalent to an electron. This process occurs when a nucleus has an excess of neutrons with respect to protons, as compared to the stable isobar. This type of transition converts a neutron into a proton; similarly, a positron is released when a proton is converted into a neutron. These decays follows the relation:
Gamma ray emission is follows the previously discussed modes of decay when the decay leaves a daughter nucleus in an excited state. This nucleus is capable of further de-excitation to a lower energy state by the release of a photon. This decay follows the relation:
Generation of electromagnetic radiation can occur whenever charged particles pass within certain distances of each other without being in fixed orbits, the accelerations (or decelerations) may give off the radiation. This is partly illustrated by the diagram at right where an electron has its course altered by near passage by a positive particle. Bremsstrahlung radiation also occurs when two electrons or other similarly charged particles pass close enough to deflect, slow down, or speed up at least one of the particles.
When high-energy radiation bombards materials, the excited atoms within emit characteristic "secondary" (or fluorescent) radiation.
A particle on the exact design trajectory (or design orbit) of the accelerator only experiences dipole field components, while particles with transverse position deviation are re-focused to the design orbit. For preliminary calculations, neglecting all fields components higher than quadrupolar, an inhomogenic Hill differential equation
can be used as an approximation, with
- a non-constant focusing force , including strong focusing and weak focusing effects
- the relative deviation from the design beam impulse
- the trajectory curvature radius , and
- the design path length ,
thus identifying the system as a parametric oscillator. Beam parameters for the accelerator can then be calculated using Ray transfer matrix analysis; e.g., a quadrupolar field is analogous to a lens in geometrical optics, having similar properties regarding beam focusing (but obeying Earnshaw's theorem).
At right is an image indicating the range of cosmic-ray energies. The flux for the lowest energies (yellow zone) is mainly attributed to solar cosmic rays, intermediate energies (blue) to galactic cosmic rays, and highest energies (purple) to extragalactic cosmic rays.
There is "a correlation between the arrival directions of cosmic rays with energy above 6 x 1019 electron volts and the positions of active galactic nuclei (AGN) lying within ~75 megaparsecs."
The Oh-My-God particle was observed on the evening of 15 October 1991 over Dugway Proving Ground, Utah. Its observation was a shock to astrophysicists, who estimated its energy to be approximately 3×1020
eV(50 joules)—in other words, a subatomic particle with kinetic energy equal to that of a baseball (142 g or 5 oz) traveling at 100 km/h (60 mph).
It was most probably a proton with a speed very close to the speed of light, so close, in fact, [(1 − 5×10−24
) × c], that in a year-long race between light and the cosmic ray, the ray would fall behind only 46 nanometers (5×10−24
light-years), or 0.15 femtoseconds (1.5×10−16
“[F]or a particle of energy E in EeV and charge Z in a magnetic field B in µG [the Larmor radius (RL)] is roughly”
- is the Larmor radius,
- is the energy of the particle in EeV
- is the charge of the particle, and
- is the constant magnetic field.
"Various gaps and density minima have been observed in the Saturnian ring system. Attempts have been made to attribute these observations to gravitational resonances with the inner satellites, thus causing the removal of particles from the dark regions of the ring system (Alexander 1953, 1962)."
"According to an alternative theory (Alfvén and Arrhenius, 1976) the observed dark regions are the result of a 'cosmogonic shadow effect' produced by the 'two-thirds fall-down mechanism'. The basis of this mechanism is the following: plasma contained in the magnetic dipole field of a central (celestial) body is brought up to a state of partial corotation. In the corotating frame of reference, the plasma experiences an outward centrifugal force which drives a current of density J.
The "charge on a dust particle changes with latitude, i.e., [going] from the equator up to the '2/3' points. The electrostatic charging of dust particles in a plasma [where] charging processes such as photo-emission, field emission thermo-ionic emission, and secondary emission can be neglected [...] contributions to the total charging current come mainly from thermal fluxes of electrons and ions, i.e., . A dust grain acquires its equilibrium charge -qe (q > 0) when"
"The electron and ion currents are given by (Mendis et al., 1984)"
"where y = -q(e2/kTC), C being the grain capacitance[, ] rg is the grain radius, the average electron density, and is the average ion (plasma) density."
"Over the last few years, the cold dark matter cosmogony has become a fiducial model for the formation of structure. [...] The problem with detecting dark matter using annihilation radiation gamma rays has been that the expected signal is comparable to the background (Stecker 1988) and it would be difficult to separate a "cosmic-ray halo" from a dark halo."
"The flux of annihilation gamma rays is given by"
"where is the cross section, is the distance to the [dwarf spheroidals] dSph's, is the [weakly interacting massive particles] WIMP's mass, is the average number of gamma rays in excess of 100 MeV per annihilation and depends weakly on (Stecker 1988)."
In March 2010 it was announced that active galactic nuclei are not responsible for most gamma-ray background radiation. Though active galactic nuclei do produce some of the gamma-ray radiation detected here on Earth, less than 30% originates from these sources. The search now is to locate the sources for the remaining 70% or so of all gamma-rays detected. Possibilities include star forming galaxies, galactic mergers, and yet-to-be explained dark matter interactions.
|violet||668–789 THz||380–450 nm|
|blue||631–668 THz||450–475 nm|
|cyan||606–630 THz||476–495 nm|
|green||526–606 THz||495–570 nm|
|yellow||508–526 THz||570–590 nm|
|orange||484–508 THz||590–620 nm|
|red||400–484 THz||620–750 nm|
The visible spectrum is the portion of the electromagnetic spectrum that is visible to (can be detected by) the human eye. Electromagnetic radiation in this range of wavelengths is called visible light or simply light. A typical human eye will respond to wavelengths from about 390 to 750 nm. In terms of frequency, this corresponds to a band in the vicinity of 400–790 THz. A light-adapted eye generally has its maximum sensitivity at around 555 nm (540 THz), in the green region of the optical spectrum (see: luminosity function).
The Sakuma–Hattori equation is a mathematical model for predicting the amount of thermal radiation, radiometric flux or radiometric power emitted from a perfect blackbody or received by a thermal radiation detector.
In its general form it looks like:
|Second Radiation Constant (0.014387752 m⋅K)|
|Temperature dependent effective wavelength in meters|
|Temperature in Kelvin.|
The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations below and the figure at right.
Here , and are the spherical coordinates of the three-dimensional vector of cartesian coordinates . is the total intensity of the beam, and is the degree of polarization. The factor of two before represents the fact that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped accompanied by a 90° rotation. The four Stokes parameters are sometimes denoted I, Q, U and V, respectively.
If given the Stokes parameters one can solve for the spherical coordinates with the following equations:
A relativistic jet coming out of the center of an active galactic nucleus is moving along AB with a velocity v. We are observing the jet from the point O. At time a light ray leaves the jet from point A and another ray leaves at time from point B. Observer at O receives the rays at time and respectively.
- , where
Apparent transverse velocity along CB,
- , where
If (i.e. when velocity of jet is close to the velocity of light) then despite the fact that . And of course means apparent transverse velocity along CB, the only velocity on sky that we can measure, is larger than the velocity of light in vacuum, i.e. the motion is apparently superluminal.
When a beam or packet of ions hits the metal it gains a small net charge while the ions are neutralized. The metal can then be discharged to measure a small current equivalent to the number of impinging ions. Essentially the faraday cup is part of a circuit where ions are the charge carriers in vacuum and the faraday cup is the interface to the solid metal where electrons act as the charge carriers (as in most circuits). By measuring the electrical current (the number of electrons flowing through the circuit per second) in the metal part of the circuit the number of charges being carried by the ions in the vacuum part of the circuit can be determined. For a continuous beam of ions (each with a single charge)
where is the number of ions observed in a time (in seconds), is the measured current (in amperes) and is the elementary charge (about 1.60 × 10−19 C). Thus, a measured current of one nanoamp (10−9 A) corresponds to about 6 billion ions striking the faraday cup each second.
In astrophysics, particularly the study of accretion disks, the epicyclic frequency is the frequency at which a radially displaced fluid parcel will oscillate. It can be referred to as a "Rayleigh discriminant". When considering an astrophysical disc with differential rotation , the epicyclic frequency is given by
where R is the radial co-ordinate. This quantity can be used to examine the 'boundaries' of an accretion disc - when becomes negative then small perturbations to the (assumed circular) orbit of a fluid parcel will become unstable, and the disc will develop an 'edge' at that point. For example, around a Schwarzchild black hole, the Innermost Stable Circular Orbit (ISCO) occurs at 3x the event horizon - at . For a Keplerian disk, .
As temperature increases in an astronomical object composed of H2, the molecules begin to dissociate.
"At a temperature of 8000 K, hydrogen gas is 99.99 percent monatomic."
where is an initial concentration [H] at low temperatures as partial particle density, is the dissociation energy 4.52 eV, k is Boltzmann's contant (8.6173324(78)×10−5 eV K-1), and T is temperature in K.
- what is the concentration of H ([H]) at T = 8000 K?
- what is [H] at T = 800 K?
- at what temperature is [H] = 1?
- what is [H] at T = 5778 K?
"Our calculations show that production of [lithium] in low-energy flares [by nucleosynthesis], taking place in the surfaces of T Tauri-like stars, is energetically possible and can give the observed excesses over the interstellar abundance."
"[T]here is evidence of lithium production in some stars due to some undefined mechanism. The observations show that the Li abundance on some red giants ... and young stars exceeds the average abundance in the universe by 2 orders of magnitude". It is "suggested that Li produced in the helium envelopes of red giants comes to the surface of the stars as the result of a strong convection." For young stars, "the production of the light elements in nonthermal nuclear reactions seems the most appropriate mechanism that can be responsible for enrichment of stellar photospheres by Li." "At least 0.3 metric tons of excited Li and Be nuclei were produced during the solar flare of 1991 November 15. One can estimate the equilibrium concentration of 7Li nuclei in the solar atmosphere by assuming that they are produced only in solar flares and that a leakage of Li nuclei occurs with the solar wind."
Although 7Be is usually assumed to have been produced by the Big Bang nuclear fusion, excesses (100x) of the isotope on the leading edge of the Long Duration Exposure Facility (LDEF) relative to the trailing edge suggest that "most of the sun's fusion must occur near the surface rather than the core." The particular reaction
3He + 4He → 7Be + γ (429 keV)
and the associated reaction chains
7Be(p,γ)8B → 2α + e+ + νe
generate 14% and 0.1% of the α-particles, respectively, and 10.7% of the present-epoch luminosity of the Sun. Usually, the 7Be produced is assumed to be deep within the core of the Sun; however, such 7Be would not escape to reach the leading edge of the LDEF.
Astrognosy deals with the materials of celestial objects and their general exterior and interior constitution.
The theoretical constitution of the Earth is illustrated using the one-dimensional Preliminary Reference Earth Model (PREM) at right. The density in kg-m-3 of radial layers is plotted against radius in km.
As the Sun consists of a plasma and is not solid, it rotates faster at its equator than at its poles. This behavior is known as differential rotation, and is caused by convection in the Sun and the movement of mass, due to steep temperature gradients from the core outwards. This mass carries a portion of the Sun’s counter-clockwise angular momentum, as viewed from the ecliptic north pole, thus redistributing the angular velocity.
Solar rotation is able to vary with latitude because the Sun is composed of a gaseous plasma. The rate of rotation is observed to be fastest at the equator (latitude φ=0 deg), and to decrease as latitude increases. The differential rotation rate is usually described by the equation:
where ω is the angular velocity in degrees per day, φ is the solar latitude and A, B, and C are constants. The values of A, B, and C differ depending on the techniques used to make the measurement, as well as the time period studied. A current set of accepted average values is:
- A= 14.713 deg/day (± 0.0491)
- B= –2.396 deg/day (± 0.188)
- C= –1.787 deg/day (± 0.253)
Def. "the radial distance q from the Sun's center such that the following finite Fourier transform is zero:
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."
R⊙eq ≈ 6.955 x 105 km. The thickness of the photosphere is about 400 km. R⊙p ≈ 6.951 x 105 km.
The density of the Sun is about 2 x 10-4 kg m-3. Or,
One mole of H2 (gas) has a mass of 2.016 x 10-3 kg. The molar density of the photosphere may be
For H2 (gas) the molar constant-volume heat capacity at 298 K is 20.18 J/(mol · K). At 2000 K it is about 25 J/(mol · K). Using a linear extrapolation,
for 5777 K, yields
Before calculating the amount of energy or power necessary to heat the coronal clouds around the Sun, let's see if the influx of electrons from outside the heliosphere may be able to heat the surface of the photosphere (p) to 5777 K from 100 K.
Voyager 1 is 17,932,000,000 km (119.9 AU) from the Sun at RA 17.163h Dec +12.44°, ecliptic latitude of 34.9°.
For this laboratory example, let the electron flux be 2 e- cm-2 s-1 diffusing into our solar system from elsewhere in the galaxy. Each of these electrons has an energy of 10 MeV.
If the electron flux measured by Voyager 1 is close to 2 e- cm-2 s-1 where each electron averages 10 MeV and these electrons are heading for the Sun, then each electron may strike the photosphere from anywhere in a sphere around the Sun.
To heat the photosphere to 5777 K takes
The power (P) that may be deposited on the photospheric surface of the Sun is
The luminosity (in Watts, W) of the Sun is 3.846 x 1026 W. In eV/s this is
If the energy of the incoming electrons is 700 MeV and the flux is 8.48 x 104 e- cm-2 s-1, then the power from the incoming electrons would be
Def. a "physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)" is called force.
In astronomy we estimate distances and times when and where possible to obtain forces and energy.
The key values to determine in both force and energy are (L/T²) and (L²/T²). Force (F) x distance (L) = energy (E), L/T² x L = L²/T². Force and energy are related to distance and time using proportionality constants.
Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:
- F is the force between the masses,
- G is the gravitational constant,
- m1 is the first mass,
- m2 is the second mass, and
- r is the distance between the centers of the masses.
In the International System of Units (SI) units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6.674×10−11
N m2 kg−2.
Observationally, we may not know the origin of the force.
Coulomb's law states that the electrostatic force experienced by a charge, at position , in the vicinity of another charge, at position , in vacuum is equal to:
where is the electric constant and is the distance between the two charges.
Coulomb's constant is
where the constant is in SI units of C2 m−2 N−1.
For reality, is the relative (dimensionless) permittivity of the substance in which the charges may exist.
The energy for this system is
where is the displacement.
Newton's second law of motion is that , where is the force applied, is the mass of the object receiving the force, and is the acceleration observed for the astronomical object. The newton is therefore:
- N: newton
- kg: kilogram
- m: metre
- s: second.
In dimensional analysis:
- M: mass
- L: length
- T: time.
But, for a force of unknown type, mass or charge may be meaningless until proven applicable.
where may be called an energy phantom, or astronomical energy phantom.
A cyclotron is a compact type of particle accelerator in which charged particles in a static magnetic field are travelling outwards from the center along a spiral path and get accelerated by radio frequency electromagnetic fields. Cyclotrons accelerate charged particle beams using a high frequency alternating voltage which is applied between two "D"-shaped electrodes (also called "dees"). An additional static magnetic field is applied in perpendicular direction to the electrode plane, enabling particles to re-encounter the accelerating voltage many times at the same phase. To achieve this, the voltage frequency must match the particle's cyclotron resonance frequency
with the relativistic mass m and its charge q. This frequency is given by equality of centripetal force and magnetic Lorentz force. The particles, injected near the center of the magnetic field, increase their kinetic energy only when recirculating through the gap between the electrodes; thus they travel outwards along a spiral path.
Cyclotron radiation is electromagnetic radiation emitted by moving charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' motion through them, creating an acceleration of charged particles that causes them to emit radiation (and to spiral around the magnetic field lines). Cyclotron radiation is emitted by all charged particles travelling through magnetic fields, however, not just those in cyclotrons. Cyclotron radiation from plasma in the interstellar medium or around black holes and other astronomical phenomena is an important source of information about distant magnetic fields. The power (energy per unit time) of the emission of each electron can be calculated using:
where E is energy, t is time, is the Thomson cross section (total, not differential), B is the magnetic field strength, V is the velocity perpendicular to the magnetic field, c is the speed of light and is the permeability of free space.
A synchrotron is a particular type of cyclic particle accelerator originating from the cyclotron in which the guiding magnetic field (bending the particles into a closed path) is time-dependent, being synchronized to a particle beam of increasing kinetic energy. The synchrotron is one of the first accelerator concepts that enable the construction of large-scale facilities, since bending, beam focusing and acceleration can be separated into different components. Unlike in a cyclotron, synchrotrons are unable to accelerate particles from zero kinetic energy; one of the obvious reasons for this is that its closed particle path would be cut by a device that emits particles. Thus, schemes were developed to inject pre-accelerated particle beams into a synchrotron. The pre-acceleration can be realized by a chain of other accelerator structures like a linac, a microtron or another synchrotron; all of these in turn need to be fed by a particle source comprising a simple high voltage power supply, typically a Cockcroft-Walton generator.
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