Blackbody Radiation and the Inverse Square Law

[ecastro]

I am currently confused with the concept of the blackbody radiation and the inverse square law.

Planck's function for the radiation of a blackbody is in $W sr^{-1} m^{-3}$, is this somehow a form of intensity (because of the watts per square meter unit)? If it does, doesn't intensity decreases with distance, and that would mean that the radiation from a blackbody differs depending on the distance of the subject? If temperature will be calculated using Planck's function, does that mean that the temperature of the object is dependent on the distance to the object?

Please help understand these concepts. Thank you in advance.

 

[soarce]

It is also defined per unit solid angle, and this removes the dependence on the distance (see wikipedia for instance). I think that although the total intensity will decrease with the distance from the source, the temperature fingerprint is the shape of spectra not the intensity.

 

[ecastro]

It is also defined per unit solid angle, and this removes the dependence on the distance (see wikipedia for instance). I think that although the total intensity will decrease with the distance from the source, the temperature fingerprint is the shape of spectra not the intensity.

How about this equation?

$I = \sigma T^4$

 

[soarce]

Ok, I misused the term "total intensity". I was thinking of the energy received by a detector calculated as the integral over the detector area, I wasn't referring to the total intensity of the source.

 

[ecastro]

I found out that the equation I just showed is luminosity. So, is the total intensity of the source is luminosity, or otherwise?

 

[soarce]

I think it is luminosity, $j(T) = \int d\nu d\Omega I(\nu,T)$ and you are left with the energy radiated per unit time (power). Further you will probably need to integrate over the source area.

 

[ecastro]

Alright. Going back to my original question, does this mean that the value of the radiation from a blackbody is independent of distance? The spectra of the Sun is the same whether here on Earth or on Mars?

 

[Drakkith]

Planck's function for the radiation of a blackbody is in Wsr−1m−3W sr^{-1} m^{-3} , is this somehow a form of intensity (because of the watts per square meter unit)? If it does, doesn't intensity decreases with distance, and that would mean that the radiation from a blackbody differs depending on the distance of the subject?

That equation describes the spectral radiance of a surface. That is, the power emitted in watts per solid angle per wavelength per square meter.

If temperature will be calculated using Planck's function, does that mean that the temperature of the object is dependent on the distance to the object?

It does not.

I found out that the equation I just showed is luminosity. So, is the total intensity of the source is luminosity, or otherwise?

As best I can tell, luminosity is a term used in astronomy to refer to the total amount of energy emitted by an astronomical object like a star or planet. The equivalent SI unit is radiant flux. Intensity doesn't seem to be used by itself, but only in conjunction with another term, such as in radiant intensity.

See the full list of units here: https://en.wikipedia.org/wiki/Radiometry

Note the explanation of irradiance in the table: Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".

 

[ecastro]

Thanks for the information. I will not be using the term intensity for awhile. Is there a relationship between the spectral radiance and the absolute square of an electric field (this is also sometimes referred as an intensity)?

 

[Drakkith]

Is there a relationship between the spectral radiance and the absolute square of an electric field (this is also sometimes referred as an intensity)?

That I can't answer. If you're referring to the amplitude of the EM waves emitted from the surface of the object, I'm sure there's a relationship, I just don't know what it is.

 

[my2cts]

A perfect blackbody radiator follows Lamberts cosine law so that the surface brightness is independent on distance or viewing angle.
In general the angular extent of the emissor is distance dependent so that the energy received becomes distance or orientation dependent
https://en.wikipedia.org/wiki/Lambertian_reflectance

 

[GTOM]

http://www.stsci.edu/jwst/instruments/nirspec/sensitivity/

Found that one about James Webb sensitivity, but i don't fully understand it.
What would be the detection range of a MJ radiation based on this data for example? What is R=100 or 1000? S/N?

 

[Drakkith]

What would be the detection range of a MJ radiation based on this data for example?

What's 'a MJ radiation'?

What is R=100 or 1000?

It refers to the resolution of the prism or grating. See this link: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/gratres.html

S/N?

I believe that refers to the signal to noise ratio. For example, the NIRSpec instrument page says this: R=100 mode shall reach a limiting continuum flux of 132 nJy at 3.0 µm at S/N=10 in t=10,000 s.

I think this means that in R=100 mode, which has less resolution but concentrates light better than the R=1000 mode, a flux of 132 nJy (nano Janskys) at a wavelength of 3 microns will generate a S/N ratio of 10 in 10,000 seconds worth of exposure time.

 

[GTOM]

Thanks. MJ radiation, i meant, a body that radiates a MJ energy in less than 10.000 second, so overall radiated energy is MJ.

What's 'a MJ radiation'?
It refers to the resolution of the prism or grating. See this link: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/gratres.html
I believe that refers to the signal to noise ratio. For example, the NIRSpec instrument page says this: R=100 mode shall reach a limiting continuum flux of 132 nJy at 3.0 µm at S/N=10 in t=10,000 s.

I think this means that in R=100 mode, which has less resolution but concentrates light better than the R=1000 mode, a flux of 132 nJy (nano Janskys) at a wavelength of 3 microns will generate a S/N ratio of 10 in 10,000 seconds worth of exposure time.

 

[Drakkith]

Thanks. MJ radiation, i meant, a body that radiates a MJ energy in less than 10.000 second, so overall radiated energy is MJ.

Oh, okay. The correct phrase should be "A MJ of energy", since a megajoule is a quantity. Like a "gallon of milk" or "meter of fabric".

What would be the detection range of a MJ radiation based on this data for example?

I don't know, honestly. Probably not very far in astronomical distances. Certainly not outside of the solar system.