翻訳と辞書 Words near each other ・ Massy, Essonne ・ Massy, Kyrgyzstan ・ Massy, Saône-et-Loire ・ Massy, Seine-Maritime ・ Massyla ・ Massyla corrugata ・ Massylii ・ Massè ・ Massé ・ Masséda ・ Massérac ・ Masséré ・ Massís de l'Ardenya ・ Mass–action ratio ・ Mass–energy equivalence ・ Mass–luminosity relation ・ Mast ・ Mast (botany) ・ Mast (film) ・ Mast (hieroglyph) ・ Mast (naval) ・ Mast (sailing) ・ Mast (Sufism) ・ MAST @ FIU Biscayne Bay Campus ・ MAST Academy ・ Mast aft rig ・ Mast Ali ・ Mast Ali, Iran ・ Mast Ali-ye Olya ・ Mast Ali-ye Sofla Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mass–luminosity relation ： ウィキペディア英語版 Mass–luminosity relation In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity. The relationship is represented by the equation: : : : :$\backslash frac)$ For stars with masses less than 0.43''M''⊙, convection is the sole energy transport process, so the relation changes significantly. For stars with masses ''M'' > 20''M''⊙ the relationship flattens out and becomes ''L'' ∝ ''M''.〔 It can be shown this change is due to an increase in radiation pressure in massive stars.〔 These equations are determined empirically by determining the mass of stars in binary systems to which the distance is known via standard parallax measurements or other techniques. After enough stars are plotted, stars will form a line on a logarithmic plot and slope of the line gives the proper value of ''a''. The mass/luminosity relation is important because it can be used to find the distance to binary systems which are too far for normal parallax measurements, using a technique called "dynamical parallax". In this technique, the masses of the two stars in a binary system are estimated, usually as being the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved.〔 The mass/luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L. One finds that more massive stars live shorter. A more sophisticated calculation factors in a star's loss of mass over time. ==Derivation== Deriving a theoretically exact mass/luminosity relation requires finding the energy generation equation and building a thermodynamic model of the inside of a star. However, the basic relation ''L'' ∝ ''M''3 can be derived using some basic physics and simplifying assumptions. The first such derivation was performed by astrophysicist Arthur Eddington in 1924. The derivation showed that stars can be approximately modelled as ideal gasses, which was a new, somewhat radical idea at the time. What follows is very similar to Eddington's approach, but uses a random-walk analysis and does not include the optical opacity. To the first approximation, stars are black body radiators with a surface area of 4''πR''2. Thus, from the Stefan–Boltzmann law, the luminosity (energy emitted per second) is obtained as :$L\; =\; 4\backslash pi\; R^2\backslash sigma\; T\_E^4,$ where ''σ'' is Stefan's constant, 5.67 × 10−8W m−2 K−4. Under the condition of hydrostatic equilibrium, :$\backslash frac=-\backslash frac$. By integrating both sides of this relation from ''r'' = 0 to ''r'' = ''R'', one arrives at a form of the virial theorem: :$\backslash langle\; P\; \backslash rangle\; =\; -\; \backslash frac\backslash frac$. The potential energy of a spherical mass distribution is :$E\_\; =-\; \backslash frac\backslash frac.$ This and the volume are substituted in to give :$\backslash langle\; P\; \backslash rangle\; \backslash approx\; \backslash frac$. Simplifying greatly, the ideal gas law (''PV'' = ''nkT'') can be used to obtain the temperature. :$\backslash langle\; P\; \backslash rangle\; =\; \backslash frackT,$ :$kT\; =\; \backslash frac$. Here $\backslash bar\; m$ is the average mass of gas particles within the star. Substituting this equation into the initial luminosity equation, along with :$R=\backslash left(\backslash frac\backslash fracM\backslash right)^\}$ yields :$L\; \backslash varpropto\; M^$. A slightly more exact result can be achieved by considering that the above equation gives the average temperature based on the average pressure, but what is actually needed is the surface temperature. Because stars are much hotter in the center than near the surface, one also needs to estimate the relationship between the surface temperature and internal temperature. The center is so much hotter since energy takes a long time to escape, otherwise, thermodynamic equilibrium would be achieved quickly and the temperature would be nearly uniform throughout. A random walk analysis can be used to estimate the "delay factor", i.e. the amount of time it takes energy to escape. In reality, the mean free path, $l$ of photons within the Sun depends on the density and temperature, but here it will be approximated as a constant. After ''N'' interactions, resulting in ''N'' vector displacements in random directions, the distance traveled is: :$\backslash mathbf$. The square of the net displacement is: :$D^2\; =\; l\_1^2\; +\; l\_2^2\; +\; \backslash cdots\; +\; l\_n^2\; +\; 2(\backslash mathbf$. Averaging over many random direction changes, the terms involving dot products cancel since the direction is random. Hence, for large $N$, :$D^2\; =\; l\_1^2\; +\; l\_2^2\; +\; \backslash cdots\; +\; l\_n^2\; =\; Nl^2.$ Thus, to escape from the Sun, on average $\backslash frac$ steps are required. The time this takes is $t\backslash approx\backslash frac$. In contrast, the time to escape directly out of the Sun from the center is $\backslash frac$, which is a factor of $\backslash frac$ shorter. Thus, factoring this into the Stefan-Boltzmann law leads to :$T\_E\backslash approx\; \backslash Big(\backslash frac\backslash Big)^\}T\_I$. Combining the above equations, the following result is obtained 〔 :$L\; \backslash approx\; 4\backslash pi\; R^2\backslash sigma\; T\_I^4\backslash frac\backslash approx\; \backslash frac\backslash fracG^4\backslash bar\; m^4\; \backslash langle\; \backslash rho\; \backslash rangle\; l\; M^3$ But the mean free path is inversely proportional to the product of the cross section and the numerical density, and thus :$l\; \backslash varpropto\; \backslash langle\backslash rho\backslash rangle^$ which gives :$L\; \backslash varpropto\; M^3.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mass–luminosity relation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.