Brightness
In particle physics, instantaneous luminosity is defined as the number of particles per unit area and per unit time in a beam. It is measured in inverse units of effective section per unit of time. By integrating this quantity over a period, the integrated luminosity is obtained, which is measured in inverse units of effective section (such as pb-1). The greater this quantity, the greater the probability that interesting events will occur in a high-energy experiment.
Given a process whose effective section, σ, we know, for a given integrated luminosity, L,, we can estimate the number of times that event will occur simply by multiplying both quantities:
- Number of events = L ×
Stellar luminosity
In astronomy, luminosity is the power (amount of energy per unit of time) emitted in all directions by a celestial body. It is directly related to the absolute magnitude of the star. This value is not constant if long enough periods are considered, since the star changes its luminosity depending on the state it is in, but it remains constant in the usual periods for humans. Although it can lead to confusion, in astronomy luminosity is a different concept from brightness; the brightness depends fundamentally on the distance at which we are from a certain object, while the luminosity is an intrinsic physical property of the star.
If the effective temperature "T" of the star's black body, the Stefan-Boltzmann Law allows us to calculate the power emitted per unit area of the star:
- PS=σ σ T4[chuckles]W/m2]{displaystyle P_{S}=sigma T^{4}quad [{text{W/}}}m^{2}]}}
![{displaystyle P_{S}=sigma T^{4}quad [{text{W/}}m^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d38d5d559a36cfb501a660d26c23fcb2e6f3f92)
Where σ σ =5.670374419⋅ ⋅ 10− − 8[chuckles]W⋅ ⋅ m− − 2⋅ ⋅ K− − 4]{displaystyle sigma =5.670374419cdot 10^{-8}quad [Wcdot m^{-2}cdot K^{-4}}}}}} is the Stefan-Boltzmann Constant
![{displaystyle sigma =5.670374419cdot 10^{-8}quad [Wcdot m^{-2}cdot K^{-4}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/511eacf1c75b6aa6c71d3a5ad8e3dc7953f8aff5)
Assuming the spherical star of radius "R" its surface area is:
- S=4π π R2[chuckles]m2]{displaystyle S=4pi R^{2}quad [m^{2}}}}}
![{displaystyle S=4pi R^{2}quad [m^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65016242053936af4ebdc87899dd9346d0d29fc)
And the brightness "L" of the star is:
- L=4π π R2σ σ T4[chuckles]W]{displaystyle L=4pi R^{2}sigma T^{4}quad [{text{W}}]
![{displaystyle L=4pi R^{2}sigma T^{4}quad [{text{W}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/557c8010e84d0344abbcf46ec86788e86fa31847)
Brightness of the Sun
The luminosity of the Sun, L☉ or LSol is the classical unit used in astronomy to compare the brightness of other stars. Its approximate value is
- L ≈ ≈ 3,827⋅ ⋅ 1026[chuckles]W]{displaystyle L_{bigodot }approx 3,827cdot 10^{26} [{text{W}}}}}}.
![{displaystyle L_{bigodot }approx 3,827cdot 10^{26} [{text{W}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29cfbb50300b0b393dc2d9d625cf3fda953bee85)
Note that this is a constant quantity, and that it does not depend on any measurement distance.
We can calculate an approximation of the constant with little data. The power density that the Earth receives from the Sun is approximately:
- P =1367[chuckles]Wm2]{displaystyle P_{bigodot }=1367~left[{frac {text{W}}}{{{{text{m}}{2}}}}}{right]}}}}.
![{displaystyle P_{bigodot }=1367~left[{frac {text{W}}{{text{m}}^{2}}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d508d6d52b79e9c6263e36f20bb60f8ecfb4b4c2)
A sphere of radius R equal to 1 AU has a surface area of
- SE=4π π R2[chuckles]m2]≈ ≈ 4⋅ ⋅ 3,1415⋅ ⋅ (1,496⋅ ⋅ 1011)2[chuckles]m2]{displaystyle S_{E}=4pi {R^{2}}~[{text{m}{2}{2}}{2}}approx 4cdot 3,1415cdot (1,496cdot 10^{11})^{2}{{text{m}}{text{2}}}}}}}}}}.
![{displaystyle S_{E}=4pi {R^{2}}~[{text{m}}^{2}]approx 4cdot 3,1415cdot (1,496cdot 10^{11})^{2}~{[{text{m}}^{2}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a854f067fca828a76c1161b2fa22942768498ca8)
- SE≈ ≈ 2,812⋅ ⋅ 1023[chuckles]m2]{displaystyle S_{E}approx 2,812cdot 10^{23}~{[{text{m}}^{2}}}}}}}{displaystyle S_{E}approx.
![{displaystyle S_{E}approx 2,812cdot 10^{23}~{[{text{m}}^{2}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c58e9e600cbe3bb3b2e47a0345e11621b5acb34)
If we assume that the power density emitted by the Sun remains constant in all directions, we can calculate the total power emitted as:
- L =P ⋅ ⋅ SE[chuckles]W]{displaystyle L_{bigodot }=P_{bigodot }cdot S_{E}~left[{text{W}}right]}}.
- L ≈ ≈ 1367[chuckles]Wm2]⋅ ⋅ 2,812⋅ ⋅ 1023[chuckles]m2]{displaystyle L_{bigodot }approx 1367~left[{frac {text{W}{{{{text{m}}{{text{2}}}}}}{right]cdot 2,812cdot 10^{23}{{{text{m}}{text{2}}}}}}}}}}}}}}}}}}}}}}}}{{{{.
- L ≈ ≈ 3,8⋅ ⋅ 1026[chuckles]W]{displaystyle L_{bigodot }approx 3,8cdot 10^{26} [{text{W}}}}}}.
![{displaystyle L_{bigodot }=P_{bigodot }cdot S_{E}~left[{text{W}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c950bcebf6de327859a81ade9c9f71add9e9e139)
![{displaystyle L_{bigodot }approx 1367~left[{frac {text{W}}{{text{m}}^{2}}}right]cdot 2,812cdot 10^{23}~{[{text{m}}^{2}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec96965373ee3265bda93e35ce23ce871e579e05)
![{displaystyle L_{bigodot }approx 3,8cdot 10^{26} [{text{W}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d466508501d3e946f4967a043f07cd35d8eec8e3)