====== Blackbody radiation ====== Blackbody radiation refers to the electromagnetic radiation emitted by a non-reflecting body held at a constant temperature. It represents a state of thermal equilibrium between matter and radiation, often realized in astrophysics within "optically thick" environments such as stellar interiors or the early universe. This radiation field is isotropic and unpolarized, and its properties depend exclusively on the temperature $T$ of the source. ===== Specific intensity ===== The fundamental descriptor of this radiation is the specific intensity, denoted as $I(\nu, T)$. This quantity measures the radiant power passing through a unit area, per unit solid angle, per unit frequency interval. For a blackbody, the specific intensity is universal and is described by the Planck function.

Temperature: 5800 K

The derivation of the blackbody spectrum relies on the Bose-Einstein distribution function, which describes the statistical behavior of integer-spin particles (bosons) such as photons. In a phase space defined by position coordinates $\mathbf{x}$ and momentum vectors $\mathbf{p}$, the distribution function $f(\mathbf{x}, \mathbf{p})$ represents the occupation number of quantum states. For photons in thermal equilibrium, this function is given by: $$f = \frac{2}{h^3} \frac{1}{e^{E/kT} - 1}$$ Here, $h$ is Planck's constant, $E = h\nu$ is the energy of the photon, and the factor of 2 arises from the two independent polarization states of the photon. The factor $h^3$ represents the volume of a single cell in six-dimensional phase space ($d^3x d^3p$). The specific intensity $I(\nu, T)$ is directly related to this phase-space density. Since photons travel at speed $c$, the intensity is the energy carried by photons passing through a surface, which equates to the product of the energy per photon $h\nu$, the phase space density $f$, the factor $c/4\pi$ related to isotropic solid angle integration, and the density of states factor $p^2/h^3$. Specifically, the Planck function relates to $f$ via the density of states in frequency space, yielding: $$I(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT} - 1}$$ This equation forms the basis for all subsequent characteristics of blackbody radiation. ===== Rayleigh-Jeans approximation ===== In the low-frequency limit, where the photon energy is much smaller than the thermal energy ($h\nu \ll kT$), the exponential term in the denominator of the Planck function can be approximated using the Taylor series expansion $e^x \approx 1 + x$. Substituting this into the specific intensity equation cancels the $h$ terms, resulting in the Rayleigh-Jeans law: $$I(\nu, T) \approx \frac{2\nu^2 kT}{c^2}$$ This approximation was historically significant as it corresponds to the classical prediction that failed at high frequencies (the "ultraviolet catastrophe"). In this regime, the intensity is directly proportional to the temperature $T$, a property extensively used in radio astronomy. Radio telescopes measure the power received and convert it into a "brightness temperature" or "antenna temperature" using this linear relation, providing a convenient metric for source intensity even for non-thermal emitters. ===== Wien approximation ===== At the opposite end of the spectrum, where frequencies are high and photon energy exceeds the thermal kinetic energy ($h\nu \gg kT$), the exponential term $e^{h\nu/kT}$ becomes very large. The $-1$ in the denominator becomes negligible, allowing the Planck function to be approximated as: $$I(\nu, T) \approx \frac{2h\nu^3}{c^2} e^{-h\nu/kT}$$ This is the Wien approximation. It describes the steep cutoff of the spectrum in the ultraviolet and X-ray regimes for typical stellar temperatures. The exponential factor dominates the $\nu^3$ term, ensuring that the total energy integrated over all frequencies remains finite, thus resolving the divergence predicted by classical physics. ===== Wien's displacement law (frequency) ===== The spectral distribution of blackbody radiation peaks at a specific frequency $\nu_{peak}$ that shifts with temperature. Differentiating the Planck function with respect to $\nu$ and solving for the maximum yields a transcendental equation, the solution of which shows that the peak frequency is linearly proportional to temperature: $$h\nu_{peak} = 2.82 kT$$ This relationship implies that hotter objects emit the bulk of their energy at higher frequencies. For example, the Cosmic Microwave Background (CMB) at 2.73 K peaks at approximately 160 GHz, whereas the Sun, at roughly 5800 K, peaks in the visible spectrum. ===== Wien's displacement law (wavelength) ===== When the specific intensity is expressed per unit wavelength interval, $I_\lambda(\lambda, T)$, the condition for the spectral peak changes due to the nonlinear relationship between frequency and wavelength differentials ($d\nu = -c/\lambda^2 d\lambda$). The peak wavelength $\lambda_{peak}$ follows the familiar form of Wien's displacement law: $$\lambda_{peak} T = 2.898 \times 10^{-3} \text{ K m}$$ It is important to recognize that the peak of the spectrum in wavelength space does not correspond to the same photon energy as the peak in frequency space. The wavelength peak occurs at a frequency roughly 1.76 times higher than the frequency peak derived from $I(\nu, T)$. ===== Stefan-Boltzmann law ===== The total radiant flux $\mathcal{F}$ emitted from the surface of a blackbody is obtained by integrating the specific intensity over all frequencies ($0 \to \infty$) and over the outward-facing hemisphere of solid angles. The integration over solid angle contributes a factor of $\pi$, and the frequency integral yields a dependence on the fourth power of temperature: $$\mathcal{F} = \sigma T^4$$ where $\sigma \approx 5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$ is the Stefan-Boltzmann constant. For a spherical star of radius $R$, this flux definition leads to the luminosity equation $L = 4\pi R^2 \sigma T^4$. This law defines the effective temperature of stars, relating their total intrinsic power output to their physical size. ===== Spectral energy density ===== The spectral energy density $u_\nu(\nu, T)$ measures the amount of radiant energy contained within a unit volume of space per unit frequency interval. For isotropic radiation, the energy density is related to the specific intensity by a factor of $4\pi/c$. This accounts for radiation traveling in all directions within the volume: $$u_\nu(\nu, T) = \frac{4\pi}{c} I(\nu, T) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu/kT} - 1}$$ This function describes the "photon gas" within a cavity. In standard cosmology, this represents the energy density of the CMB photons that permeate the universe. ===== Total energy density ===== Integrating the spectral energy density over the entire frequency range yields the total energy density $u(T)$, representing the total joules of radiation energy per cubic meter. Like the flux, this quantity scales with the fourth power of the temperature: $$u(T) = a T^4$$ The radiation constant $a$ is related to the Stefan-Boltzmann constant by $a = 4\sigma/c$. This relationship is critical in cosmology and stellar interior physics, as it governs the energy content of the radiation field which can dominate over matter energy density at sufficiently high temperatures. ===== Spectral number density ===== The spectral number density $n_\nu(\nu, T)$ defines the number of photons per unit volume per unit frequency interval. It is derived by dividing the spectral energy density $u_\nu$ by the energy of a single photon, $h\nu$: $$n_\nu(\nu, T) = \frac{u_\nu(\nu, T)}{h\nu} = \frac{8\pi \nu^2}{c^3} \frac{1}{e^{h\nu/kT} - 1}$$ Physically, this can be interpreted as the product of the density of quantum states in phase space (proportional to $\nu^2$) and the Bose-Einstein occupation number (the exponential term). This distribution determines the number of photons available to interact with matter at a specific energy. ===== Total number density ===== The total number density of photons $n(T)$ is found by integrating $n_\nu$ over all frequencies. The result depends on the third power of the temperature, rather than the fourth: $$n(T) \propto T^3$$ Specifically, $n(T) \approx 2.03 \times 10^7 T^3 \text{ photons m}^{-3}$. This scaling has important cosmological implications; as the universe expands and temperature drops, the number density of photons decreases effectively as $1/R^3$ (where $R$ is the scale factor), conserving the total number of photons in a comoving volume, whereas the energy density drops as $1/R^4$ due to the additional redshift of photon energy. ===== Average photon energy ===== The average energy of a photon in a blackbody radiation field is the ratio of the total energy density to the total number density ($u/n$). Since $u \propto T^4$ and $n \propto T^3$, the average energy is directly proportional to $T$: $$h\nu_{avg} \approx 2.70 kT$$ This value provides a useful rule of thumb for characteristic photon energies. It is slightly lower than the energy at the frequency peak ($2.82 kT$) but provides a representative energy for interactions such as ionization or scattering within a thermal plasma. ===== Inquiries ===== - Demonstrate how the Planck function reduces to the Rayleigh-Jeans approximation in the low-frequency limit ($h\nu \ll kT$) using the Taylor series expansion $e^x \approx 1+x$. - Explain why the spectral peak in frequency space ($\nu_{peak}$) corresponds to a different photon energy than the spectral peak in wavelength space ($\lambda_{peak}$), specifically referencing the relationship between the differentials $d\nu$ and $d\lambda$. - Using the scaling relations provided, determine the factor by which the total luminosity of a spherical star increases if its temperature doubles while its radius remains constant. - Contrast the temperature dependence of the total photon energy density ($u \propto T^4$) with that of the total photon number density ($n \propto T^3$) and explain what this implies for the average energy per photon as temperature changes. - State the equation of state relating radiation pressure $P$ to total energy density $u$ for an isotropic blackbody field and identify the factor distinguishing it from the kinetic pressure of a non-relativistic gas.