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Main forms

Planck's law expresses the quantity of radiation emitted by a black body, or ideal radiator, as a function of the absolute temperature T of the radiator and the frequency $ν$ of the portion of radiation being so expressed. It is customarily expressed as

B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}}.

Roughly speaking, $B ν (T)$ gives the power in watts radiated normally from unit area A of radiator into unit solid angle Ω within a band of width unit frequency centered on frequency $ν$ . More precisely, $B ν (T) d A dΩ dν$ gives the power in watts radiated normally from infinitesimal area dA into infinitesimal solid angle dΩ in a frequency band of infinitesimal width $dν$ .

The quantity $B ν (T)$ can thus be seen to be a form of power density called spectral radiance, namely radiated power divided by area, solid angle, and frequency bandwidth. When all parameters are given in SI units, spectral radiance has units $W.m -2 .sr -1 .Hz -1$ , watts per square meter per steradian per Hertz.

Radiation not normal to the surface is treated as follows. A black body is by definition a Lambertian radiator. That is, radiation from area A at an angle θ to the normal to that area is treated as coming from the smaller area $A cos(θ)$ resulting from projecting A onto a plane normal to the radiation, which can then be treated as though it were a normally radiating black body. It follows that the total radiation at any given frequency emitted normally to a black-body disk equals the total radiation of that frequency emitted from a spherical black body of the same temperature and radius in the same direction; indeed the two objects produce indistinguishable beams in that direction. (The Sun is not perfectly Lambertian, and hence not quite a black body, on account of limb darkening.)

Planck's law can also be expressed as a function of wavelength $λ=c/ν$ instead of $ν$ . Substituting $c/λ$ for $ν$ in the expression for $B ν (T)$ yields

{\frac {2hc}{\lambda ^{3}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}.

This would be fine as long as we continued to use $dν$ as the differential when integrating. What we really want however is a law $B λ (T)$ that we can integrate over wavelength in the homogeneous form $B λ (T) dλ$ rather than the mixed form $B λ (T) dν$ . We should therefore have started not with just $B ν (T)$ but $B ν (T) dν$ , and substituted for both $ν$ and $dν$ .

Now the derivative of $ν=c/λ$ is $dν/dλ=-c/λ 2$ . Ignoring the minus sign and substituting $c/λ 2 dλ$ for $dν$ then yields the wavelength form of Planck's law, namely

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

The minus sign is accounted for by noticing that integration from a small to a large frequency becomes from a large to a small wavelength, and interchanging the wavelength limits so as to give the usual increasing order absorbs the sign.

The integrals of Planck's law

Planck's law can be integrated over each of area, solid angle, and bandwidth.

Area Planck's law is independent of the geometry, whence integrating $B ν (T) d A$ over a planar area A simply gives $B ν (T) A$ as the spectral intensity of the total area in watts per steradian per Hertz in the direction normal to that plane. The total spectral intensity of a spherical black body is trickier on account of Lambert's law. Here we can use the equivalence noted above between a disk and a sphere: the spectral intensity of a unit hemisphere of area 2π in the direction normal to its base equals that of the base itself, which has area π, which is what we should take for A in the formula $B ν (T) A$ for spectral intensity. Had we naively taken A to be the area of the hemisphere we would have overestimated its spectral intensity by a factor of two.

Solid angle Integrating $B ν (T) dΩ$ over the whole hemisphere into which an infinitesimal surface element dA radiates gives the spectral radiant exitance for that surface element. This can be analyzed by treating it as just the inverse of the area law in the case of a spherical radiator, turned inside out and viewed as a spherical receiver of radiation from its center. Again a factor of two is lost, and the spectral radiant exitance is therefore simply $π B ν (T)$ in watts per square meter per Hertz. Since the only quantitative difference between spectral radiance and spectral radiant exitance is a factor of π, Planck's law can conveniently be stated in either form as it is a triviality to convert to the other form.

Bandwidth In general integrating Planck's law over an arbitrary range of frequencies is best accomplished by a suitable numerical integration algorithm. Over the entire spectrum however, that is, from DC (zero Hertz) to infinity, the integration can be performed analytically. Doing this for spectral radiant exitance yields the Stefan-Boltzmann law in units of watts per square meter.

Other units

Other units besides Hertz for frequency and meters for wavelength are in common use. Changing frequency units to other frequency units is simpler than switching between frequency and wavelength because $ν$ is simply multiplied by a constant. Nevertheless $dν$ participates even in this simpler situation, obliging us to substitute in $B ν (T) dν$ rather than just $B ν (T)$ as before, and likewise with the wavelength form.

The dimensions of $dν$ being the same as for $ν$ , and likewise for $λ$ , a scale factor of s can be seen to become s⁴ for $B ν (T) dν$ and s⁻⁴ for $B λ (T) dλ$ , as one might expect given the Stefan-Boltzmann law's fourth-power dependence. For example spectroscopists commonly prefer CGS units over MKS, and in place of frequency $ν$ use wavenumber $ν̃ = 1/λ$ cm⁻¹ (so $ν = cν̃$ ). In this case $B ν (T) dν$ must be scaled by a factor of c⁴, giving

2hc^{2}{\tilde {\nu }}^{3}{\frac {1}{e^{hc{\tilde {\nu }}/(k_{\mathrm {B} }T)}-1}}

as the "spectroscopist's version" of Planck's law, with the units now being watts per square meter (still MKS) per steradian per cm⁻¹. (For watts per cm² divide by 10,000.)

Occasionally Planck's law is expressed in terms of angular frequency $ω$ radians per second, related to frequency by {{math|ν = ω/2π}. The scaling factor in that case becomes (2π)⁻⁴, giving

B_{\omega }(T)={\frac {\omega ^{3}}{8\pi ^{4}c^{2}}}{\frac {1}{e^{\hbar \omega /(k_{\mathrm {B} }T)}-1}}.

However it is customary in that case to also replace h by 2πℏ, so that the wavelength formula then becomes

B_{\omega }(T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}{\frac {1}{e^{\hbar \omega /(k_{\mathrm {B} }T)}-1}}

How Planck's law unifies the Wien and Rayleigh-Jeans laws

The details of how Planck actually arrived at his law are recounted in the history section below. Here we put the Wien and Rayleigh-Jeans laws in forms that make it clear how to pass directly from them to Planck's law with a simple trick.

The Wien approximation is

B_{\nu }(T)=a{\frac {\nu ^{3}}{e^{b\nu /T}}}

where a and b can be observed empirically to be about 1.5E-50 and 4.8E-11 respectively at sufficiently high frequencies. The Rayleigh-Jeans law is

B_{\nu }(T)=d\nu ^{2}T.

where d is observed empirically to be 3.1E-40 at sufficiently low frequencies. Both are in units of watts per square meter per steradian per Hertz, which is equivalent dimensionally to joules per square meter (steradians have no dimension and Hertz have dimension sec⁻¹).

Now we know from the Wien displacement law that the frequencies comprising the black body curve, viewed as a fixed shape, increase linearly with temperature. It is therefore natural to introduce a quantity $ν/ T$ , call it g for now, whose possible values (in Hertz per degree) express the points on the curve in a temperature-independent way. (In particular the wavelength peak of the Wien displacement law is at g = 1.03E11.) We can then rewrite the Wien approximation and Rayleigh-Jeans law as respectively

B_{\nu }(T)=1.5E-50{\frac {\nu ^{3}}{''e''^{4.8E-11''g''}}}

and

B_{\nu }(T)=3.1E-40{\frac {\nu ^{3}}{g}}.

Already they look more similar. We also know that these laws work well in respectively the ultraviolet and infrared ends of the black body curve but blow up in the ends where they don't work. Now for small g the denominator e^c₂g of Wien's approximation tends to 1 + 4.8E-11 g. Simply subtracting 1 would make it tend instead to 4.8E-11 g, bringing it into line with Rayleigh-Jeans while remaining correct for large g. This gives us a unified law

B_{\nu }(T)=1.5E-50{\frac {\nu ^{3}}{e^{4.8E-11\nu /T}-1}}.

We also have Boltzmann's formula k_BT for the energy of two degrees of freedom of a molecule. We therefore introduce a new constant h = 4.8E-11 k_B = 6.6×10⁻³⁴ so as to make the exponent $h ν/ k B T$ . In order for subtracting 1 to make sense dimensionally, the quantity $h ν$ must have the same dimension as k_BT, namely energy. (Note that we did not assume that $h ν$ is an energy term, the logic of the situation forced this conclusion!)

Since the units for this law are joules per square meter, the quantity $1.5E-50 \nu3$ must be an energy divided by an area. We make an inspired guess at $hν$ for the energy and $λ 2$ for the area. The quotient $hν/λ 2$ equals $hν 3 /c 2$ , and the coefficient $h/c 2$ evaluates to about 0.75E-50 or half of the empirically observed coefficient 1.5E-50. To achieve that coefficient therefore requires introducing a factor of 2, giving as the final form of Planck's law

{\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}},

which of course is Planck's law, albeit arrived at by inspiration rather than perspiration.

References

Bibliography

Proof (truth)

For the notion of proof as a measure or indicator of the quality of some product or service see Proof (quality).

A proof is sufficient evidence for the truth of a proposition. In oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a perlocutionary speech act intended to persuade a hearer or reader of the truth of a proposition. In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and other previously established theorems. Logic, in particular proof theory, formalizes and studies the notion of proof. In epistemology, jurisprudence, and theology, the notion of justification plays approximately the role of proof.

Proofs need not be verbal. The ancient Greeks took the shadow cast on the moon by the earth during a lunar eclipse as proof that the earth was round. Suitably incriminating evidence left at the scene of a crime may serve as proof of the identity of the perpetrator.

Conversely a verbal entity need not assert a proposition to constitute a proof of that proposition. A signature constitutes direct proof of authorship; less directly, handwriting analysis may be submitted as proof of authorship of a document. Evidence of plagiarism is usually sufficient as proof of nonauthorship. Privileged information in a document can serve as proof that the document's author had access to that information; such access might in turn establish the location of the author at certain time, which might then provide the author with an alibi.

In general there is no absolute threshold of sufficiency at which evidence becomes proof. The same evidence that may convince one jury may not persuade another. An experienced mathematician may find a short demonstration of a theorem sufficient where a novice would need more clarification. Proof theory provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious."

Proof (quality)

Proofreading, reviewing a manuscript or artwork for errors or improvements
Prepress proof, a facsimile of press artwork for job verification
Artist's proof, a single print taken during the printmaking process
Proof coinage, a coin made as an example of a particular strike
Proof test, for a firearm
Homeopathic proving, a homeopathic procedure
Galley proof, a preliminary version of a publication
Alcoholic proof, a measure of an alcoholic drink's strength

Ordinal number

In mathematics, an ordinal number, or just ordinal, is a transitive set of ordinals, or hereditarily transitive set. That is, every element of an ordinal is transitive and its elements in turn are transitive and so on down. The Axiom of regularity stops this recursion after finitely many steps, namely at the empty set.

The finite ordinals are 0 = {}, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, …. This is a consequence of two remarkable facts, that every element of a finite transitive set is itself a finite transitive set (so we can drop "ordinal" as a condition for membership in a finite ordinal), and only one set of each finite cardinality can be transitive, whence the finite ordinals can be identified with their cardinalities as per the foregoing enumeration.

It is evident for the finite ordinals that the successor of an ordinal α is α∪{α}. In fact this holds for all ordinals. We write α∪{α} as α+1.

The set of all finite ordinals is transitive and hence itself an ordinal. It is the least infinite ordinal, and is denoted ω. It is the canonical example of a countable set, and its cardinality is denoted $\aleph _{0}$ .

Like the finite ordinals, ω has a successor ω+1, namely {0,1,2,…,ω}. That in turn has a successor {0,1,2,…,ω,ω+1}, and we can continue in this way until we reach {0,1,2,…,ω,ω+1,ω+2…} = ω+ω, denoted ω·2. Continuing faster we eventually reach ω·2+ω = ω·3. We could consider ω·3+1, but moving yet faster we arrive at ω·4, ω·5, and eventually ω·ω, denoted ω². Picking up yet more speed, we arrive at ω³, ω⁴, … and then ω^ω. Getting into high gear we speed past ω^ω, ω^{ω^ω}, ω^{ω^{ω^ω}}, …, to arrive at ε₀, the least ordinal α satisfying α = ω^α.

Throughout this entire sequence, every ordinal has been countable, that is, its cardinality has been $\aleph _{0}$ . However one cannot infer from this that every countable ordinal is ω. To see this, consider listing all infinitely many even numbers first, and then continue by listing the odd numbers. Then 1 is preceded only by even numbers, but there is no even number immediately before 1 because there is no largest even number. So unlike the usual enumeration of the natural numbers, in which every number but 0 has a predecessor, this enumeration is structurally different, that is, it is not order isomorphic to ω. Instead it is order isomorphic to ω+ω = ω·2, because the set of even numbers standardly ordered is order isomorphic to ω, and likewise the odd numbers.

Furthermore we have not come close to exhausting the countable ordinals: ε₀+1 is countable, and so on. To get past the countable ordinals requires a new insight.

Every ordinal is linearly ordered by inclusion, for example 2 < 4 because {0,1} is a subset of {0,1,2,3}. This linear order is in fact a well-order, that is, every nonempty set of ordinals has a least element. Hence for every predicate that is not false of all ordinals, there exists a least ordinal satisfying that predicate. Furthermore no two ordinals are order isomorphic, and every well-ordered set is order isomorphic to some ordinal, whence the ordinals can serve to encode the order types of all well-ordered sets however large.

There exist uncountable ordinals, whence by the foregoing there exists a least uncountable ordinal, denoted ω₁. The cardinality of this ordinal is denoted $\aleph _{1}$ , which is the next cardinal after $\aleph _{0}$ . There is a sequence of increasingly large ordinals as measured by their cardinality, denoted ω₁, ω₂, ω₃, …, with corresponding cardinalities $\aleph _{1}$ , $\aleph _{2}$ , $\aleph _{3}$ , …. The cardinality of an ordinal defines a many to one association from ordinals to cardinals, with all ordinals between ω_i inclusive and ω_i+1 exclusive having the same cardinality $\aleph _{i}$ .

Eigenvalue etc.

In mathematics, the notion of eigenvalue is defined in the context of an algebraically closed field F, typically the complex numbers, a vector space V over F, and a linear map f: V → V thereon, typically represented as an n×n matrix when V is of finite dimension n. An eigenvalue of such a linear map f is a vector v for which there exists a scalar λ (an element of F) satisfying f(v) = λv.

An eigenvector of a linear map may be visualized as a vector that can be "stretched" by the map but whose direction cannot be changed by it. When the eigenvector's associated eigenvalue is real, "stretching" has its usual intuitive meaning of simply changing the length of the vector. For example the matrix [[2,.3],[0,.2]] has an eigenvector [1,0] with eigenvalue 2 and its action on that eigenvector is simply to double its length, while that of [[.5,.3],[0,.2]], which has the same eigenvector, is to halve it.

A matrix such as [[0,-1],[1,0]] on the other hand may have complex eigenvectors and associated eigenvalues even though the matrix itself has only real entries. One eigenvalue of this matrix is [i,1], with associated eigenvalue i, and the action of this matrix on that eigenvector is to rotate it 90 degrees counterclockwise. Another eigenvector of the same matrix is [1,i] with eigenvalue −i, which this matrix rotates 90 degrees clockwise. This rotational action of a matrix on such an eigenvector furnishes it with an oscillating character independently of any ordinary stretching that it may also induce.

Eigenvectors provide an elegant way of analyzing the long-term behavior of linear systems whose evolution is described by a linear map f. Consider a linear system with n parameters understood as a vector space V of finite dimension n. Let v be some vector in V representable as a sum v = v₁ + … + v_n of eigenvectors. By linearity the action of f is to transform v into the vector λ₁v₁ + … + λ₁v_n. Applying f k times in this way therefore transforms v into λ₁^kv₁ + … + λ_n^kv_n.

This long-term behavior can be further extended to the limiting behavior of the system by letting k tend to infinity. In this limiting case, when |λ_i| < |λ_j|, the contribution of λ_i^kv_i becomes vanishingly small compared to that of λ_j^kv_j. Hence in the limit the only eigenvectors that matter are those for which the magnitude |λ| of their eigenvalue λ is maximal.

As a case in point the pair [F_k+1,F_k] of consecutive Fibonacci numbers as a vector can be understood as the result of multiplying the vector [1,0] by the matrix [[1,1],[1,0]] k times. This matrix has eigenvectors [φ,1] and [ψ,1] with associated eigenvalues φ and ψ where φ = (1+√5)/2 = 1.618… and ψ = 1−φ = −0.618…. We can represent the initial vector [1,0] as the sum [φ,1]/√5 − [ψ,1]/√5, whence F_k = (φ^k − ψ^k)/√5, for example F₇ = (1.618…⁷ − (−.618…)⁷)/√5 = 13. In the limit ψ^k becomes negligible and F_k becomes essentially just φ^k.

An eigenspace of f: V → V is any subspace of V spanned by eigenvectors of f having the same eigenvalue λ. Every vector of an eigenspace is "stretched" and/or "wobbled" in the same way by f, in the sense that applying f k times to any v in that eigenspace yields λ^kv.

Eigenvectors are crucial to the concept of equilibrium of a linear system with n parameters understood as one evolving under the action of an n×n matrix. A linear system that has attained equilibrium in this sense must be in a state that is a linear combination of eigenvectors whose eigenvalues all have the same magnitude. These eigenvalues all lie on a common circle in the complex plane, such that all other eigenvalues of the matrix lie properly within that circle. There need be no relationship between the locations of those maximal-magnitude eigenvalues around the circle, whence the limiting behavior of a linear system with n parameters may have as many as n different and unrelated frequencies in its oscillatory behavior.

Suggested greenhouse effect lead

The greenhouse effect is the heating of a planet's atmosphere resulting from the atmosphere being more transparent to incoming radiation from the planet's hot sun than to the outgoing radiation from the planet as warmed by that sun. Earth's atmosphere is heated in this way due primarily to the presence of water vapor (0.4% average) and secondarily to carbon dioxide (0.04%), called greenhouse gases. The principal components of Earth's atmosphere, nitrogen and oxygen, are transparent to both incoming and outgoing radiation and are not considered greenhouse gases. In contrast Venus's atmosphere is largely carbon dioxide (97%), and moreover is two orders of magnitude more massive than Earth's atmosphere.

This heating of the atmosphere by the outgoing radiation further warms the planet's surface above its effective temperature, the temperature the planet would theoretically achieve by direct radiation in the absence of greenhouse gases but taking reflection or albedo into account. Earth's effective temperature is −18 °C, which its greenhouse gases raise to 15° C. Venus is closer to the Sun but has a much higher albedo than Earth, with the result that its effective temperature is −46 °C, which its massive carbon dioxide atmosphere raises to over 460 °C.