Radiative energy transport refers to the transfer of energy through electromagnetic waves or photons. It occurs as a result of the emission and absorption of electromagnetic radiation by matter, and is the primary mechanism for transferring energy in the form of heat or light in many physical systems.

In general, radiative energy transport can occur through three basic mechanisms: blackbody radiation, thermal radiation, and fluorescence. Blackbody radiation refers to the emission of electromagnetic radiation from a body at a temperature above absolute zero, and it is described by Planck's law. Thermal radiation, on the other hand, refers to the emission of electromagnetic radiation from matter at any temperature due to the excitation of its constituent atoms and molecules. Finally, fluorescence is a special case of thermal radiation where the energy is absorbed and then re-emitted at a different wavelength.

Here are a few key points about radiative energy transport in astrophysics:

Energy generation: In many celestial objects, energy is generated through processes like nuclear fusion in the cores of stars, and this energy is transported to the surface through radiative energy transfer.
Energy transfer in stars: In stars, radiative energy transfer occurs through the emission and absorption of electromagnetic radiation by the gas and dust that make up the star's interior. The energy is transferred from the hot core to the cooler outer regions, where it is eventually emitted as light.
Energy balance: Radiative energy transfer plays an important role in maintaining the energy balance in celestial objects. For example, in a star, the rate of energy generation must be balanced by the rate of energy transfer to the surface to prevent the star from collapsing or expanding.
Star formation: Radiative energy transfer also plays a role in the formation of stars. For example, the heat generated by gravitational collapse during the formation of a star can be transported to the surface through radiative energy transfer, preventing the star from becoming too hot and disrupting the formation process.
Energy transfer in galaxies: In galaxies, radiative energy transfer occurs through the emission and absorption of electromagnetic radiation by the gas, dust, and stars that make up the galaxy. This energy transfer plays a role in the dynamics and evolution of galaxies, and is an important factor in determining the distribution of stars and other celestial objects within a galaxy.

In conclusion, radiative energy transport is a key aspect of astrophysics, and plays a crucial role in the behaviour and evolution of celestial objects.

Let us find radiative energy transport in Astrophysics. A photon can travel relatively far during its random walk without being absorbed or dispersed if the star has a low opacity (or low density) or a low mass absorption coefficient. In this situation, energy is transported by means of radiative process. We know that the relation between luminosity and flux density is given by

L = 4πr² F

L = 4πr² [ Energy / Area x time ]

L = 4πr² [Force / Area] [distance/time ] ......i

Let us consider a spherical shell of thickness dr and mass dMr at the radius r. The distance travelled by radiation per unit time in the medium is expressed in terms of optical thickness at the medium, i.e

distance / time = c / 𝜏

In free space 𝜏 -> 1, i.e velocity of radiation approaches to c. Thus equation 1 can be written as

L = 4πr² [pressure] [ c / 𝜏 ]

As is well known, local thermodynamic equilibrium is observed in the thin spherical shell of gases, .ie gases are in perfect equilibrium with radiation. In this situation, Planck's law should apply here with the appropriate gas distribution for a thin, spherical shell with thickness dr,

Lr / c = 4πr² dp/ d𝜏 ........iii

The optical thickness is defined as

d𝜏 = α dr ......iv

Here, α is the opacity of the medium. This term says how effectively medium can obscure radiation.

Since opacity depends on the wavelength of radiation, the average opacity can be expressed in terms of Planck's function (or specific intensity) if there is local thermodynamic equilibrium. This opacity is termed as Rosseland mean density and is given by

Thus, eq iii takes a form

Lr / c = 4πr² dP/ α dr [ Using iv]

In the case of LTE, the expression takes a form,

The Rosseland mean opacity can be expresses in terms of density as

here χ is uniform opacity. Thus equation v takes the form,

The radiation pressure is given by
Here a is the radiation constant. Thus vii becomes

This is our required expression for radiative temperature gradient. To solve this expression, we should know density profile and temperature profile only.

The radiative temperature gradient is an important aspect of astrophysics and has several advantages including:

Energy transfer in stars: The radiative temperature gradient plays a crucial role in energy transfer in stars, as it determines how energy generated in the core of a star is transported to the surface. The radiative temperature gradient helps to maintain the energy balance in a star, ensuring that the rate of energy generation is balanced by the rate of energy transfer to the surface.
Star formation: The radiative temperature gradient also plays a role in star formation, as it helps to dissipate heat generated by gravitational collapse during the formation process. This helps to prevent the formation of stars that are too hot, which could disrupt the formation process.
Understanding of physical processes: The radiative temperature gradient can be used to study physical processes like energy transfer and heat dissipation in stars, providing valuable insights into the behavior and evolution of these objects.
Characterization of exoplanets: The radiative temperature gradient can also be used to study exoplanets, or planets outside of our solar system. By analyzing the radiative temperature gradient of exoplanets, astronomers can determine their atmospheric conditions and study their climate and weather patterns.
Energy transfer in galaxies: The radiative temperature gradient also plays a role in energy transfer in galaxies, as it determines how energy is transferred between the stars, gas, and dust that make up a galaxy. This helps to determine the distribution of stars and other celestial objects within a galaxy.

This note is taken from Astrophysics, MSC Nepal.

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