#### Stellar Atmospheres, Physics tutorial

Introduction:

The atmosphere of a star is stated as such layers adequately close to the surface in such a way that a few photons can escape from the star's surface. In another words; the atmosphere is as deep as one can spot into the star.

Classification of Stellar Atmosphere:

The main layers in a stellar atmosphere are as follows:

1) Photosphere:  In this, energy transfer is dominated via radiation methods. The photosphere of Sun is 400 km thick and has temperature more than 5800K.

2) Chromospheres: Non-irradiative energy dissipation heats up the gas to above the irradiative equilibrium temperature; though, the density is adequately high that most of the dissipated energy can be radiated away and the resulting temperature is not too high. The Sun's chromosphere is 104 km thick and consists of temperature equivalent to 1.5 × 104 K.

3) Corona: Significant non-irradiative energy transport and dissipation; the density is too low for the dissipated energy to be proficiently radiated away; as a result the corona is extremely hot.

4) Stellar wind: The outmost layer of a stellar atmosphere is so tenuous that it is not longer gravitationally bound; most stars, comprising our Sun, lose mass via a stellar wind.

Model of the Stellar Atmosphere:

The theory of stellar atmospheres comprises constructing models of the photosphere which take into account the effective temperature, gravity and the elemental profusions.

In modeling the given suppositions are employed for the stellar atmosphere:

1) Atmosphere is spherically symmetric.

2) Elements mixture is homogeneous by depth.

3) Atmosphere is in the hydrostatic equilibrium.

4) Atmosphere is time-independent.

5) The mass of atmosphere is small as compared by the total stellar mass.

6) There are no sources or sink of energy.

7) Energy transport occurs via radiation and convection (that is, no heat conduction)

Construction of a Model Stellar Atmosphere:

To form a more realistic model atmosphere, it is essential to comprise the supposition of hydrostatic equilibrium represented as:

dP/dZ = -ρg

That is, represented as: dP/dτv = dPg/dτs + dPr/dτs = ρg/kvs

Here, 'vs' is a fixed standard frequency and 'τv' and 'τs' is the optical depth at that frequency (that is, computed for μ = 1) and Pg and Pr are the gas and radiation pressures correspondingly.

If Pr << Ps (valid if given star is not too hot, or too opaque and consists of a high surface gravity) then,

dPg/dτs ≈ ρg/Kvs

By using the equation of state of an ideal gas Pg = NKT

Here, 'N' is the number density of gas particles, 'μ' is the mean molecular weight and mB is the atomic mass of the hydrogen.

From above, we can equate the above equations as follows:

Pg = NKT = ρgKT/μmB

By replacing this in above, we get:

dP/dτv = μmB Pg/KτKs

[lnPg/dτv = μmBg/KτKvs]

This can now be employed to prescribed variations in Pg(vs)for a particular distribution of the temperature.

Standard Solar Model

The Standard Solar Model (or SSM) refers to the mathematical treatment of the Sun as the spherical ball of gas (in varying states of ionization, having the hydrogen in the deep interior being fully ionized plasma). This model, that is technically spherical symmetric quasi-static model of a star, consists of stellar structure illustrated through several differential equations derived from the fundamental physical principles. The model is constrained through boundary conditions, namely the luminosity, radius, age and composition of the Sun, that are well determined.

The composition in the photosphere of the modern-day Sun, via mass, is around 74.9% hydrogen and 23.8% helium. All heavier elements, termed as metals in astronomy, account for less than 2 % of the mass. The SSM is employed to test the validity of stellar evolution theory. Though, the only way to determine the two free parameters of the stellar evolution model, the helium abundance and the mixing length parameter (employed to model convection in the Sun), are to adjust the SSM to fit the observed Sun.

The SSM fulfills two aims:

1) It gives estimates for the helium abundance and mixing the length parameter by forcing the stellar model to contain the correct luminosity and radius at the age of Sun.

2) It gives a way to assess more complex models with additional physics, like rotation, magnetic fields and diffusion or improvements to the treatment of convection, like modeling turbulence, and convective overshooting.

Similar to the Standard Model of particle physics and the standard cosmology model, the SSM changes over time in response to the relevant latest theoretical or experimental physics discoveries.

As illustrated above, the Sun consists of an irradiative core and a convective outer envelope. In the core, the luminosity due to nuclear reactions is transmitted to the outer layers principally through radiation. Though, in the outer layers the temperature gradient is so huge that the radiation can't transport adequate energy. As a result, thermal convection takes place as thermal columns carry hot material to the surface (that is, photosphere) of the Sun. Once the material cools off at the surface, it plunges back downward to the base of the convection zone, to obtain more heat from the top of the irradiative zone.

Construction of a Solar Model:

In the construction of a solar model, as illustrated in stellar structure, one requires to consider the given parameters: temperature T(r), the density ρ(r), total pressure (matter plus radiation) P(r), luminosity l(r) and energy generation rate per unit mass ε(r) in a spherical shell of a thickness dr at a distance r from the centre of the star. The irradiative transport of energy is illustrated by the irradiative temperature gradient equation:

dT/dr = - 3Kρl/64πr2σT3

Here 'k' is the opacity of the matter, 'σ' is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.

Convection is illustrated by using mixing length theory and the corresponding temperature gradient equation (for the adiabatic convection) is:

dT/dr = (1 - 1/γ)(T/P) (dP/dr)

Here, γ = Cp/Cv is the adiabatic index, the ratio of specific heats in the gas. (For a completely ionized ideal gas, γ =5/3)

Near the base of the Sun's convection zone, the convection is adiabatic, however close to the surface of the Sun, convection is non-adiabatic.

The solar radiation explains the visible and near-visible (that is, ultraviolet and near-infrared) radiation emitted from the sun. The different areas are illustrated by their wavelength range in the broad band range of 0.20 to 4.0 µm (microns). Terrestrial radiation is a term employed to illustrate infrared radiation emitted from the atmosphere.

The given is a list of the components of solar and terrestrial radiation and their approximate wavelength ranges:

 Ultraviolet 0.20-0.39 µm Visible 0.39-0.78 µm Near-Infrared 0.78-4.00 µm Infrared 4.00-100.00 µm

Around 99% of the solar, or short-wave, radiation at the earth's surface is contained in the region from 0.3 to 3.0 µm whereas most of terrestrial, or long-wave, radiation is contained in the region from 3.5 to 50 um.

Outside the atmosphere of earth, solar radiation consists of an intensity of around 1370 W/m2. This is the value at mean earth-sun distance at the top of the atmosphere and is termed to as the Solar Constant. On the surface of the earth on a clear day, at noon, the direct beam radiation will be around 1000W/m2 for numerous locations.

The availability of energy is influenced by location (comprising latitude and elevation), season, and time of day. All of which can be readily determined. Though, the biggest factors influencing the available energy are cloud cover and other meteorological conditions which differ by location and time.

To compute the amount of sunlight reaching the ground, both the elliptical orbit of the Earth and the attenuation via the atmosphere of Earth has to be taken into account. The extraterrestrial solar Illuminance (Eext), corrected for the elliptical orbit by employing the day number of the year (dn), is represented by:

Eext = Esc [1 + 0.033412 cos (2π (dn - 3)/365)]

Here, dn =1 on January 1; dn = 2 on January 2; dn= 32 on February 1 and so on.

In this formula dn-3 is employed, as in modern times, the perihelion of Earth, the closest approach to the Sun and thus the maximum Eext take place around January 3 each year. The value of 0.033412 is find out knowing that the ratio between the perihelion (0.98328989 AU) squared and the aphelion (1.01671033 AU) squared must be roughly 0.935338.

The solar Illuminance constant (Esc), is equivalent to 1.28 × 105 Lx. The direct normal Illuminance (Edn), corrected for the attenuating effects of the atmosphere is represented by:

Edn = Eext e-cm

Here, 'c' is the atmospheric extinction coefficient and 'm' is the relative optical air mass.

Solar Constant:

The solar constant is basically a measure of the flux density. This is the amount of incoming solar electromagnetic radiation per unit area which would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit (AU) (approximately the mean distance from the Sun to the Earth). The 'solar constant' comprises all kinds of solar radiation, not just the visible light. Its value was assumed to be an average of around 1.366 kW/m2. This value differs slightly with solar activity, however recent recalibrations of the relevant satellite observations point out a value closer to 1.361 kW/m2 is much more realistic.

Intensity in the Solar System:

Different bodies of the Solar-System obtain the light of intensity inversely proportional to the square of their distance from Sun.

The real brightness of sunlight which would be noticed at the surface depends as well on the presence and composition of the atmosphere. For illustration, the thick atmosphere Venus reflects more than 60% of the solar light it obtains. The real illumination of the surface is approximately 14,000 lux, comparable to that on Earth in the daytime with overcast clouds. This describes why sunlight on Mars would be more or less like daylight on Earth wearing sunglasses and can be seen in the pictures taken via the unmanned Rover planetary probes. Therefore it would provide a perception and feel very much similar to the daylight of Earth.

A rough table comparing the quantity of solar radiation obtained by each and every planet in the Solar System is as shown:

 Planet Perihelion - Aphelion Distance (AU) Solar radiation maximum and minimum (W/m2) Mercury 0.3075 - 0.4667 14,446 - 6,272 Venus 0.7184 - 0.7282 2,647 - 2,576 Earth 0.9833 - 1.017 1,413 - 1,321 Mars 1.382 - 1.666 715 - 492 Jupiter 4.950 - 5.458 55.8 - 45.9 Saturn 9.048 - 10.12 16.7 - 13.4 Uranus 18.38 - 20.08 4.04 - 3.39 Neptune 29.77 - 30.44 1.54 - 1.47

For the comparison aim, sunlight on Saturn is slightly brighter than the sunlight of Earth at the average sunset or sunrise. Even on Pluto the sunlight would still be bright adequate to nearly match the average living room. To view sunlight as dim as full moonlight on the Earth, a distance of around 500 AU (~ 69 light-hours) is required.

The spectrum of the solar radiation of Sun is close to that of a black body by a temperature of around 5,800 K. The Sun emits EM radiation across most of the electromagnetic spectrum as illustrated below. However the Sun generates Gamma rays as an outcome of the nuclear fusion method, these super high energy photons are transformed to lower energy photons before they reach the surface of sun and are emitted out into space, so the Sun does not give off any gamma rays to speak of.

The Sun does, though, emit X-rays, ultraviolet, infrared, visible light and even Radio waves. If ultraviolet radiation is not absorbed via the atmosphere or other protective coating, it can cause damage to the skin termed as sunburn or trigger an adaptive change in the human skin pigmentation.

The spectrum of electromagnetic radiation striking the atmosphere of Earth spans a range of 100 nm to around 1 mm. This can be categorized into five regions in rising order of wavelengths.

1) Ultraviolet C or (UVC) range that spans a range of around 100 to 280 nm. The word ultraviolet refers to the fact that the radiation is at higher frequency than the violet light (and therefore as well invisible to the human eye). Owing to the absorption via the atmosphere very little reaches the surface of Earth (Lithosphere). This spectrum of radiation comprises of germicidal properties, and is employed in the germicidal lamps.

2) Ultraviolet B or (UVB) range covers 280 to 315 nm. It is as well greatly absorbed via the atmosphere, and all along by UVC is responsible for the photochemical reaction leading to the generation of the ozone layer.

3) Ultraviolet A or (UVA) covers 315 to 400 nm. It has been usually held as less damaging to the DNA, and therefore employed in tanning and PUVA therapy for psoriasis.

4) Visible range or light covers 380 to 780 nm. As the name proposes, it is this range that is visible to the bare eye.

5) Infrared range which covers 700 nm to 106 nm (1 mm). This is responsible for a significant portion of the electromagnetic radiation that reaches the Earth.

Surface Illumination:

The spectrum of surface illumination based on the solar elevation due to the atmospheric effects, having the blue spectral component from atmospheric scatter dominating throughout twilight before and after sunrise and sunset, correspondingly, and red dominating throughout sunrise and sunset. Such effects are evident in the natural light photography where the principal source of illumination is sunlight as mediated via the atmosphere. The preferential absorption of sunlight through ozone over long horizon paths provides the zenith sky its blueness whenever the sun is close to the horizon.

Climate Effects:

On Earth, the solar radiation is of course as daylight whenever the sun is above the horizon. This is for the period of daytime, and as well in summer close to the poles at night, however not at all in winter close to the poles. If the direct radiation is not blocked via clouds, this is experienced as sunshine, joining the perception of bright white light (that is, sunlight in the strict sense) and warming. The warming on the body, the ground and other objects based on the absorption (that is, electromagnetic radiation) of the electromagnetic radiation in the form of heat.

The quantity of radiation intercepted through a planetary body differs inversely by the square of the distance between the star and the planet. The orbit of Earth and obliquity change by time (over thousands of years), at times making a nearly perfect circle, and at other times stretching out to the orbital eccentricity of 5% (presently 1.67%). The net insolation remains approximately constant due to Kepler's second law, represented as:

2A/r2 dt = dθ

Here, 'A' is an invariant representing the 'real velocity'. This signifies that the integration over the orbital period (as well invariant) is a constant.

oT 2A/r2 dt = o dθ = constant

If we suppose the solar radiation power 'P' as a constant over time and the solar irradiation represented by the inverse-square law, we can get the average insolation as a constant.

The seasonal and latitudinal distribution and intensity of the solar radiation obtained at the surface of Earth varies. For illustration, at latitudes of 65 degrees the change in solar energy in summer and winter can differ by more than 25% as an outcome of the orbital variation of the Earth. As changes in summer and winter tend to offset, the change in the annual average insolation at any particular location is close to zero, however the redistribution of energy between summer and winter does strongly influence the intensity of seasonal cycles. These changes related by the redistribution of the solar energy are considered a probable cause for the coming and going of latest ice ages.

Sunlight or solar radiation is all the electromagnetic radiation given off by the Sun. Here on Earth, our atmosphere filters the light of sun, protecting us from injurious radiation and changing the color of sunlight.

At first, let us look at where this radiation comes from. As you perhaps know, the intense pressures and temperature at the core of the Sun is where the magic of solar fusion occurs. Protons are transformed into helium atoms at a rate of 600 million tons per second. As the output of this method has lower energy than the protons which began, the fusion gives off a great amount of energy in the form of gamma rays. Such gamma rays are absorbed through particles in the Sun, and then re-emitted. Over the course of 200,000 years, photons of light make their journey via the irradiative zone of the Sun and out into the space.

The surface of the Sun we can view is termed as the photosphere, and it is the point at which the light from the Sun can finally escape into the space. Through their long journey via the Sun, though, the photons have lost energy, and become other wavelengths of the light. That is a good thing; or else, we had just contained gamma rays streaming from the Sun.

The Solar radiation is not any one type of light. It is in reality a mixture of different wavelengths. The heat we sense from the sun is infrared, and ranges in wavelength from 1400 nm to 1 mm. visible light than ranges from 400 to 700 nm. Out in space, the light of sun appears white, however here on Earth we see it as more yellow as the atmosphere deflects blue and violet photons more simply. We are as well strike by ultraviolet radiation; luckily, much of this is absorbed via the atmosphere of Earth as it is quite hazardous to life.

The entire life on Earth based on solar radiation. This is the primary source of energy to the Earth, and drives the weather of planet and ocean circulation. Devoid of this source of energy, the Earth would freeze, and only its internal geothermic heat would stop it from freezing to a solid rock.

Our Relationship with the Sun:

As human beings, we have a tendency to have a love-hate relationship with the sun - on one hand; sunlight keeps us warm, produces food and shelter for us through plant life, and provides us light. On the other hand, as greenhouse gases trap more heat and the ozone layer lets more dangerous UV radiation in, the sun's rays can be noticeably dangerous. UV rays causes skin cancer in animals and humans, however can contrastingly recover other skin conditions such as psoriasis. We require the sun biologically, and also; as it cause our bodies to make vital vitamin D.

The Solar radiation and sunlight make it possible for the Earth to house life. The negative features of our relationship with the sun are mainly the result of human irresponsibility: we increase skin cancer if we ignore our bodies' signals to avoid sunlight, and we struggle by global warming as we have ignored the ecological concerns of our actions. If we do not give solar radiation the respect it deserves, we are exactly playing with fire.

Measuring the radiation of sun's output consists of a variety of useful purposes: it lets for the growth of solar energy devices, makes it possible for doctors to issue advice regarding sun exposure, and permits scientists to expect rates of future global warming on a grander scale.

Measurement Techniques:

The measurement of solar radiation is mainly based on the rate of kilowatts per square meter, represented as W/m2. This is the measurement standard for the scientific data, designed for producing a direct estimate of solar  energy - fundamentally, how much sunlight is striking a specific portion of the Earth at any given time. Measurements taken for the aim of energy production through solar panels and other photovoltaic equipment might be computed in kilowatt-hours per square meter, or kWh/m2, designed to represent the quantity of energy being produced by that sunlight.

Some of the different tools can be employed to measure kilowatts and kilowatt-hours per square meter. These comprise:

1) Pyranometer that is a tool used to assess the global solar radiation. This is comprised of a thermopile sensor having a black coating that absorbs all solar radiation, and a glass dome, that limits the spectral response of the thermopile.

2) Pyrheliometer that computes the direct radiation. A Pyrheliometer works likewise, however is designed by a solar tracker to keep the machine directly aimed at the sun for the period of the measurement being taken.

The Black Body is any object which is a perfect emitter and a perfect absorber of radiation. Object doesn't have to appear black before it can be termed as a black body. The surfaces of sun and earth behave approximately similar to black bodies. Therefore, let us define some fundamental black-body radiation laws:

Stefan-Boltzmann Law:

The Stefan-Boltzmann law associates the net amount of radiation emitted via an object to its temperature:

E = σT4

Here:

E = Net amount of radiation emitted via an object per square meter (Watts m-2)

σ is the Constant termed as the Stefan-Boltzmann constant

Stefan-Boltzmann constant = 5.67 x 10-8 Watts m-2 K-4

T = the temperature of the object in K

Now, take the Earth and Sun for instance:

For the Sun: T = 6000 K so E = 5.67 x 10-8 Watts m-2K-4(6000 K)4 = 7.3 x 107 Watts m-2

Weins Law

This law associates the wavelength of solar radiation to its surface temperature. Most of the objects emit radiation at lots of wavelengths. Though, there is one wavelength where an object emits the biggest amount of radiation represented as Wien's wavelength and written as λ:

λmax = 2897m/T(K)

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