(For all of these problems, where necessary assume the normal ratio of total- to-selective extinction.) Suppose a dust grain has an albedo \(a(\lambda)\). It is near a star whose spectrum is that of a blackbody at temperature \(T_{*}\), and whose radius is \(R\) - The grain is a distance \(d\) from the star. Derive an expression for the dust temperature. (You will have to leave your answer in terms of an integral, since the form of \(a(\lambda)\) is not given.)

In physics, blackbody radiation refers to the type of electromagnetic radiation that is emitted by a body in thermal equilibrium. The defining characteristic of a blackbody is that it absorbs all incident radiation, regardless of wavelength or angle of incidence. Consequently, it emits radiation with a spectrum that only depends on its temperature. This makes blackbody radiation an ideal concept for understanding how objects, including stars, emit energy.

For a star, which can be approximated as a blackbody, its spectrum can be characterized by blackbody radiation at a specific temperature, denoted as \( T_* \). The radiation it emits follows the Planck function, which provides the intensity of radiation as a function of wavelength and temperature. This radiation gets absorbed by surrounding dust grains, leading to changes in their temperature.

Thermal equilibrium is a state where the energy entering a system equals the energy leaving it. For a dust grain near a star, this means the energy it absorbs from the star's radiation must balance the energy it emits.
When the dust grain reaches this equilibrium, its temperature will remain stable. The incoming energy from the star's radiation is absorbed based on the grain's albedo, \( a(\lambda) \), which is a measure of how much light the grain reflects versus absorbs. Conversely, the energy that the dust grain emits is dependent on its own temperature and surface area, which follows the Stefan-Boltzmann law.

The Planck function describes the distribution of radiation emitted by a blackbody in thermal equilibrium at a given temperature. It is crucial for determining the energy absorbed by the dust grain. The function is given by:

\[ B_{\lambda}(T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1} \]

where \( B_{\lambda}(T) \) is the spectral radiance at wavelength \( \lambda \) and temperature \( T \), \( h \) is Planck’s constant, \( c \) is the speed of light, and \( k \) is Boltzmann’s constant.

Integrating this function over all wavelengths gives the total energy radiated by the star. For the dust grain absorbing this radiation, we adjust the Planck function by the grain's albedo \( a(\lambda) \) to find the absorbed energy.

The Stefan-Boltzmann law is a principle in physics that states that the total energy radiated per unit surface area of a blackbody is directly proportional to the fourth power of the blackbody's temperature. Mathematically, it is expressed as:


\[ E = \sigma T^4 \]
where \( E \) is the radiated energy per unit area, \( T \) is the absolute temperature, and \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \text{ W m}^{-2} \text{K}^{-4} \)).

In the context of a dust grain near a star, the law helps calculate the energy the dust grain emits at temperature \( T_d \). By equating this emitted energy with the absorbed energy, we can derive an expression for the dust grain temperature in thermal equilibrium.