Astronomy

What units are used for the Stefan-Boltzmann law?

What units are used for the Stefan-Boltzmann law?


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I have a star with given temperature in Kelvin and radius in solar radii. I tried to calculate the luminosity of the star using Stefan Boltzmann's law, and got an absurd number (over 1 million). What am I doing wrong, and are there any units that I should use instead of Kelvin and solar units?


The Stefan-Boltzmann constant $sigma$ is not a dimensionless quantity, it comes with units. So whatever units you use, you must ensure that the value you use for the Stefan-Boltzmann constant is consistent with them.

So using the value expressed in terms of SI units:

$$sigma = 5.670,374,419ldots imes 10^{-8}, m W,m^{-2},K^{-4}$$

you would either have to work with radius, luminosity and temperature in metres, watts and kelvins, or convert $sigma$ to the units you are actually using.

For example, if you want to work in terms of solar radii and luminosities you have to account for the conversion factors $L_odot = 3.828 imes 10^{26}, m W$ and $R_odot = 6.957 imes 10^8, m m$, giving:

$$sigma = 7.169ldots imes 10^{-17} L_odot, R_odot^{-2}, m K^{-4}$$


What is Stefan–Boltzmann Law – Stefan-Boltzmann Constant – Definition

Radiation heat transfer rate, q [W/m 2 ], from a body (e.g. a black body) to its surroundings is proportional to the fourth power of the absolute temperature and can be expressed by the following equation:

where σ is a fundamental physical constant called the Stefan–Boltzmann constant, which is equal to 5.6697×10 -8 W/m 2 K 4 . The Stefan–Boltzmann constant is named after Josef Stefan (who discovered the Stefa-Boltzman law experimentally in 1879) and Ludwig Boltzmann (who derived it theoretically soon after). As can be seen, radiation heat transfer is important at very high temperatures and in a vacuum.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m 2 ). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 0.4 to 0.76 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

Q = εσA1-2(T 41 −T 42) [J/s]

q = εσ(T 41 −T 42) [J/m 2s]

The area factor A1-2, is the area viewed by body 2 of body 1, and can become fairly difficult to calculate.


B.3 Complex Exponentials

A complex exponential e i ⁢ ϕ , where i 2 = - 1 and ϕ is any dimensionless real variable, is a complex number in which the real and imaginary parts are sines and cosines given by Euler’s formula

e i ⁢ ϕ = cos ϕ + i sin ϕ . (B.3)

Euler’s formula can be derived from the Taylor series

cos ⁡ ϕ = 1 - ϕ 2 2 ! + ϕ 4 4 ! - ϕ 6 6 ! + ⋯ ,
sin ⁡ ϕ = ϕ - ϕ 3 3 ! + ϕ 5 5 ! - ϕ 7 7 ! + ⋯ ,
e ϕ = 1 + ϕ + ϕ 2 2 ! + ϕ 3 3 ! + ϕ 4 4 ! + ⋯ .

e i ⁢ ϕ = 1 + i ⁢ ϕ - ϕ 2 2 ! - i ⁢ ϕ 3 3 ! + ϕ 4 4 ! + i ⁢ ϕ 5 5 ! - i ⁢ ϕ 6 6 ! - i ⁢ ϕ 7 7 ! + ⋯
= ( 1 - ϕ 2 2 ! + ϕ 4 4 ! - ϕ 6 6 ! + ⋯ ) + i ⁢ ( ϕ - ϕ 3 3 ! + ϕ 5 5 ! - ϕ 7 7 ! + ⋯ )
= cos ⁡ ϕ + i ⁢ sin ⁡ ϕ .

Complex exponentials (or sines and cosines) are widely used to represent periodic functions in physics for the following reasons:

They comprise a complete and orthogonal set of periodic functions. This set of functions can be used to approximate any piecewise continuous function, and they are the basis of Fourier transforms (Appendix A.1 ).

They are eigenfunctions of the differential operator—that is, the derivatives of complex exponentials are themselves complex exponentials:

d ⁢ e i ⁢ ϕ d ⁢ ϕ = i ⁢ e i ⁢ ϕ , d 2 ⁢ e i ⁢ ϕ d ⁢ ϕ 2 = - e i ⁢ ϕ , d 3 ⁢ e i ⁢ ϕ d ⁢ ϕ 3 = - i ⁢ e i ⁢ ϕ , d 4 ⁢ e i ⁢ ϕ d ⁢ ϕ 4 = e i ⁢ ϕ , … .

Most physical systems obey linear differential equations, a low-pass filter consisting of a resistor and a capacitor, for example. A sinusoidal input signal will yield a sinusoidal output signal of the same frequency (but not necessarily with the same amplitude and phase), while a square-wave input will not yield a square-wave output. The response to a square-wave input can be calculated by treating the input square wave as a sum of sinusoidal waves, and the filter output is the sum of these filtered sinusoids. This is the reason why periodic waves or oscillations are almost always treated as combinations of complex exponentials (or sines and cosines).

Real periodic signals can be expressed as the real parts of complex exponentials:

cos ⁡ ϕ = Re ⁢ ( e i ⁢ ϕ ) ,
sin ⁡ ϕ = Im ⁢ ( e i ⁢ ϕ ) .

Adding and subtracting the equations

e i ⁢ ϕ = cos ⁡ ϕ + i ⁢ sin ⁡ ϕ ,
e - i ⁢ ϕ = cos ⁡ ϕ - i ⁢ sin ⁡ ϕ

cos ⁡ ϕ = e i ⁢ ϕ + e - i ⁢ ϕ 2 (B.4)

sin ϕ = e i ⁢ ϕ - e - i ⁢ ϕ 2 ⁢ i . (B.5)

The advantage of complex exponentials over the equivalent sums of sines and cosines is that they are easier to manipulate mathematically. For example, you can use complex exponentials to calculate the output spectrum of a square-law detector (Section 3.6.2 ) without having to remember trigonometric identities. A square-law detector is a nonlinear device whose output voltage is the square of its input voltage. If the input voltage is cos ⁡ ( ω ⁢ t ) , the output voltage is

cos 2 ⁡ ( ω ⁢ t ) = ( e i ⁢ ω ⁢ t + e - i ⁢ ω ⁢ t 2 ) 2
= e 2 ⁢ i ⁢ ω ⁢ t + 2 + e - 2 ⁢ i ⁢ ω ⁢ t 4
= 2 ⁢ cos ⁡ ( 2 ⁢ ω ⁢ t ) + 2 4
= 1 2 ⁢ [ cos ⁡ ( 2 ⁢ ω ⁢ t ) + 1 ] .

The output spectrum has two frequency components: one at twice the input frequency ω and the other at zero frequency (DC).


Examples

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun's surface. He learned from the data of Charles Soret (1854&ndash1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a warmed metal lamella. A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater, namely 29 × 3/2 = 43.5.

Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.57 4 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of a lamella, so Stefan got a value of 5430 °C or 5700 K (modern value is 5780 K). This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C were claimed. The lower value of 1800 °C was determined by Claude Servais Mathias Pouillet (1790-1868) in 1838 using the Dulong-Petit law. Pouilett also took just half the value of the Sun's correct energy flux. Perhaps this result reminded Stefan that the Dulong-Petit law could break down at large temperatures.

Temperature of stars

The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation. [1] [2] So:

where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature. This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:

where , is the solar radius, and so forth.

With the Stefan–Boltzmann law, astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes in so called Hawking radiation.

Effective temperature of the Earth

Similarly we can calculate the effective temperature of the Earth TE by equating the energy received from the Sun and the energy transmitted by the Earth:

where TS is the temperature of the Sun, rS the radius of the Sun, and a0 is the distance between the Earth and the Sun. Thus resulting in an effective temperature of 6°C on the surface of the Earth.

In summary, the surface of the Sun is 21 times as hot as that of the Earth taken as a blackbody, and therefore it emits 190,000 times as much energy per square meter. The distance from the Sun to the Earth is 215 times the radius of the Sun, reducing the energy per square meter by a factor 46,000. Taking into account that the cross-section of a sphere is one fourth of its surface area, we see that there is equilibrium of approximately 342 W per m 2 surface area, or 1,370 W per m 2 cross-sectional area.

The above derivation is a rough approximation only as it assumes the Earth is a perfect blackbody. If we include the effect of Terrestrial albedo which is about 30% (meaning the actual amount of solar energy absorbed by our planet is 70% of top of atmosphere irradiation), the above equation gives an average Earth surface temperature of 255 K only. The "missing" 33 K between such calculated value and the actual measured one (288 K) is deemed to be the result of greenhouse gases, namely water vapor, carbon dioxide and methane [3] . However such a reasoning has to be taken very carefully as the Stefan–Boltzmann law is not directly applicable to non-blackbody objects variation in albedo (absorptivity or emissivity) as a function of wavelength, for example as due to greenhouse gases, can change the resulting equilibrium temperature substantially.


Summary

James Clerk Maxwell showed that whenever charged particles change their motion, as they do in every atom and molecule, they give off waves of energy. Light is one form of this electromagnetic radiation. The wavelength of light determines the color of visible radiation. Wavelength (λ) is related to frequency (f) and the speed of light (c) by the equation c = λf. Electromagnetic radiation sometimes behaves like waves, but at other times, it behaves as if it were a particle—a little packet of energy, called a photon. The apparent brightness of a source of electromagnetic energy decreases with increasing distance from that source in proportion to the square of the distance—a relationship known as the inverse square law.

5.2 The Electromagnetic Spectrum

The electromagnetic spectrum consists of gamma rays, X-rays, ultraviolet radiation, visible light, infrared, and radio radiation. Many of these wavelengths cannot penetrate the layers of Earth’s atmosphere and must be observed from space, whereas others—such as visible light, FM radio and TV—can penetrate to Earth’s surface. The emission of electromagnetic radiation is intimately connected to the temperature of the source. The higher the temperature of an idealized emitter of electromagnetic radiation, the shorter is the wavelength at which the maximum amount of radiation is emitted. The mathematical equation describing this relationship is known as Wien’s law: λmax = (3 × 10 6 )/T. The total power emitted per square meter increases with increasing temperature. The relationship between emitted energy flux and temperature is known as the Stefan-Boltzmann law: F = σT 4 .

5.3 Spectroscopy in Astronomy

A spectrometer is a device that forms a spectrum, often utilizing the phenomenon of dispersion. The light from an astronomical source can consist of a continuous spectrum, an emission (bright line) spectrum, or an absorption (dark line) spectrum. Because each element leaves its spectral signature in the pattern of lines we observe, spectral analyses reveal the composition of the Sun and stars.

5.4 The Structure of the Atom

Atoms consist of a nucleus containing one or more positively charged protons. All atoms except hydrogen can also contain one or more neutrons in the nucleus. Negatively charged electrons orbit the nucleus. The number of protons defines an element (hydrogen has one proton, helium has two, and so on) of the atom. Nuclei with the same number of protons but different numbers of neutrons are different isotopes of the same element. In the Bohr model of the atom, electrons on permitted orbits (or energy levels) don’t give off any electromagnetic radiation. But when electrons go from lower levels to higher ones, they must absorb a photon of just the right energy, and when they go from higher levels to lower ones, they give off a photon of just the right energy. The energy of a photon is connected to the frequency of the electromagnetic wave it represents by Planck’s formula, E = hf.

5.5 Formation of Spectral Lines

When electrons move from a higher energy level to a lower one, photons are emitted, and an emission line can be seen in the spectrum. Absorption lines are seen when electrons absorb photons and move to higher energy levels. Since each atom has its own characteristic set of energy levels, each is associated with a unique pattern of spectral lines. This allows astronomers to determine what elements are present in the stars and in the clouds of gas and dust among the stars. An atom in its lowest energy level is in the ground state. If an electron is in an orbit other than the least energetic one possible, the atom is said to be excited. If an atom has lost one or more electrons, it is called an ion and is said to be ionized. The spectra of different ions look different and can tell astronomers about the temperatures of the sources they are observing.

5.6 The Doppler Effect

If an atom is moving toward us when an electron changes orbits and produces a spectral line, we see that line shifted slightly toward the blue of its normal wavelength in a spectrum. If the atom is moving away, we see the line shifted toward the red. This shift is known as the Doppler effect and can be used to measure the radial velocities of distant objects.


Stefan’s Law In Astrophysics

As we have already read, Stefan’s law was the first formula with which we estimated the temperature of the Sun. Not only the Sun, Stefan’s law can be used to calculate the surface temperature of the stars too. Once we know the luminosity and dimensions of the star, we can plug in the values and find the temperature. This formula for luminosity is also useful in calculating the stellar masses of galaxies, provided we know the luminosity of the Sun accurately (which we do!). Once we know the stellar mass of the galaxy, we can find its specific star formation rate too.

Stefan’s law is not very popular but it is a very important relation in astrophysics. It can be derived from thermodynamics and also from the Planck’s law. In the previous article, we saw how spectroscopy and atomic physics were at play in astrophysics. Today’s article gives the glimpse of the importance of thermodynamics in astrophysics.


THE STEFAN-BOLTZMANN LAW


The Stefan-Boltzmann Law allows us to determine how much energy comes from a given area, say 1 square meter, of an object that emits a continuous spectrum. How much energy is emitted from this given area depends only on the temperature of the object! All objects (be it silver, iron, lead) that produce a continuous spectrum when heated will emit the same amount of energy if they have the same temperature. If we can determine the temperature for an object (maybe using Wien's Law?), we can then determine how much energy is emitted from each square meter of the object.

Note: The emitted energy goes as T (temperature) to the 4th power! So if I make T (the temperature) 4 times larger, the energy emitted from each square meter increases by 4 x 4 x 4 x 4 = 256 times!


Understanding the Stefan-Boltzmann Law (when the surroundings are hotter)

Summary: 1.My book tells me that given ##T_##, and ##T## of the object radiating heat, the law is expressed as ##H = sigma A (T^4 - T^4_)##.

2. Relating Newton's Law of Cooling, Conduction, and Stefan-Boltzmann Law

3. Is emissivity the same as Stefan's constant or is it ## e * sigma## where ##e## varies, depending on the material?

2. Stefan-Boltzmann Law is formulated as ##H = Asigma T^4## where ##H## is the energy emitted per unit time, ##A## is the area of the object, ##T## is the absolute temperature of the object and (3.) I am unclear about whether ##sigma## represents emissivity or ##e*sigma## represents Stefan's constant.

My book also defines Conduction (as the time rate of heat flow for a given temperature difference), as ##H = kA frac ## where ##H## is the rate of flow of heat (heat current), ##A## is the area of cross-section and ##L## is the length between the two points being considered, ##T_c - T_d## the temperature difference between the points.

Newton's Law of Cooling is stated as a special case of Stefan-Boltzmann Law where the temperature difference is very small and is formulated as ##frac

= k(T_2 - T_1)##.

I feel that the three must be somehow interrelated, am I correct in assuming this? If so, how?

3. Lastly is emissivity the same as Stefan's constant or is it ## e * sigma## where ##e## varies, depending on the material?


What Is the Application of Stefan-Boltzmann Law?

According to Teach Astronomy, the Stefan-Boltzmann Law can be applied to a star's size in relation to its temperature and luminosity. It can also apply to any object emitting a thermal spectrum, including metal burners on electric stoves and filaments in light bulbs.

According to Hyper Physics, the Stefan-Boltzmann Law states that the thermal energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature. The law is also related to the energy density in the radiation in a given volume of space.

According to Teach Astronomy, the mathematical form of the Stefan-Boltzmann Law states that the luminosity of a star is proportional to the star's surface area and the fourth power of its surface temperature. Therefore, changing the temperature or radius of a star changes the amount of energy radiated, or luminosity. This is why hotter stars radiate bluer light and more light per unit area at every wavelength than cooler stars. The law is used to calculate the radii of stars. The Stefan-Boltzmann Law can also be seen in everyday occurrences. For example, when an iron poker is heated, it goes from glowing red to glowing yellow as the temperature rises.


Astronomy college course/Introduction to stellar measurements

The figure to the left shows what Wikipedia calls the w:Cosmic distance ladder. [4] The ladder analogy arises because no one technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distances at the next higher rung.

    Luminosity: In astronomy, luminosity is total amount of energy emitted by a star or other astronomical object per unit time. In SI units this is expressed as joules per second or watts. I.e. The Sun has a total power output of 3.846×10 26 Watts (that's a lot of light bulbs!). A more convenient unit of luminosity is this Sun itself: 1.00 solar luminosity, or 1 L ⊙ ≈ 3.85 × 10 26 W approx 3.85 imes 10^<26>,W,> .

    : The change of angular position of a star as seen from Earth, due to the motion of the Earth around the Sun. The star's angular shift that occurs when the observer moves by AU (one astronomical unit) In 1989 the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motnios allowing measurements of stellar parallax for stars up to about 500 parsecs away, a little more than one percent of the diameter of the Milky Way Galaxy. [3]
  • Standard candle: an astronomical object that has a known luminosity.

Angular size, size, and distance Edit

  • s = r θ r a d ≈ r θ d e g 57.3 >approx r >><57.3>>,> is the arclength of a portion of a circle of radius r described the angle θ. The two forms allow θ to be measured in either degrees or radians (2π rad = 360 deg). The lengths r and s must be measured in the same units.

Stellar parallax Edit

  • D p a r s e c = b A U θ a r c s e c >= >>< heta _>>>> , where D is the distance to the object in parsecs, θ is the parallax angle in arcseconds, and b is the baseline in AU b=1 for observations taken from Earth. One degree is 60 arcminutes and one arcminute is 60 arseconds. One AU ≈ 1.5x10 11 meters, and one parsec ≈ 3.26 light-years, and one light-year ≈ 9.5×10 15 meters.

Newton's version of Kepler's third law Edit

Kepler (1571-1630) found a relation between period and average distance for all planets and comets around the Sun. Newton (1643-1727) discovered "universal" laws which not only explained Kepler's third law, but showed that the applied on earth, around other planets, as well as for stars and clusters of stars. His addition of the "total mass" allows us to "weigh" (technically "mass" almost anything and everything in the universe.

Normalized units Edit

Kepler's third law is a relation between the period of a planet and an averaged distance (semi-major axis) from the Sun. It takes on a simple form if the period is measured in years and the distance is measured in AU. For Earth, we have:

In contrast, if time is measured in seconds and distance in meters, then the Kepler's third law for Earth looks like this:

In the previous section, the Kepler/Newton relation between mass, period and semi-major axis are most conveniently written using the following units:

  • Time is measured in years
  • Distance is measured in AU
  • Mass is measured in solar masses.

In the next sections, we find it helpful to define:

  • Power (L) is measured in units of the Sun's power output
  • Temperature (T) is measured in units of the Sun's temperature.
  • Radius (R) is measured in units of the Sun's radius.

To remind the reader that these normalized, a tilde shall be placed over the variables. If a star's temperature is expressed as T

Two facts about how blackbodies "glow" Edit

A hot object gives off electromagnetic radiation. We sometimes experience this as visible light given off by every hot objects, or as heat given off by a campfire. While these phenomena have been observed for centuries, a mathematical understanding only emerged only by the first decade of the 20th century with the revolutionary idea that light was both a particle ('photon') and a wave. The following formulas are strictly true only for a blackbody, but they are good approximations for stars, since most photons that strike a star get 'lost' in the star. The color of a black body is closely related to it's temperature. Here 'color' refers to the peak wavelength emitted by the star:


Watch the video: What Units are used in Physics? (May 2022).