Brightness Temperature, Thermal Spectrum

In summary, the conversation discusses a problem that shows how astronomers can be misled into thinking that a cosmic source of radiation is thermal when it is not. The problem involves a compact source emitting one hundred solar luminosities at a frequency of 22 GHz, and the question asks what temperature an astronomer would infer if they matched the spectrum of this source with a thermal spectrum. The relevant equations include the Planck function for blackbody radiation and the flux equation. One possible approach to solving the problem is to use Wien's displacement law, which relates the peak frequency in a blackbody spectrum to its temperature.
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jmm5872
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This is a problem intended to show us that Astronomers are sometimes fooled into thinking that a cosmic source of radiation is thermal, when it is not. Here is the problem:

Consider a cosmic source emitting one hundred solar luminosities all within 1 MHz of the single frequency of 22 GHz, and assume that the source is compact; only 0.001 parsecs in radius. Such a source exists in NGC 3079. What crazy temperature would an Astronomer infer by matching the spectrum of this source at 22 GHz with a thermal spectrum?

Relevant equations:

Planck function for blackbody radiation (for frequency) B = (2hv3/c2)*1/(e(hv/kT)-1)

h = Planck constant
v = frequency
c = speed of light
k = Boltzmann constant
T = temp

flux: F = L/4(pi)r2

First I solved the Planck function for T, and I have all the values I need to plug in except B (blackbody specific intensity, which is erg s-1 cm-2 Hz-1 sr-1) So my idea to get B was to find the flux and manipulate until I have B.

First I calculated the total luminosity...100(solar luminosity) = 3.8e35 erg s-1 s-1

Then the flux, using r = 0.001pc = 3.0857e15 cm and got: 3175.896 erg s-1 cm-2

Since this is energy per unit time per unit area, and I need energy per unit time per unit area per unit frequency per unit steradian, I divided the flux above by the given frequency (2.2e10 Hz), then divided again by 2pi, which is one hemisphere of steradians. This should give me everything I need.

From calculations B = 2.297e-8 erg s-1 cm-2 Hz-1 sr-1

Then I plugged in B, and the rest of the values to calculate T, but the answer was negative.

I hope my explanation makes sense, it is hard to explain on here.
I'm not even sure if my approach was correct, if so I'm not sure if I calculated B correctly.

Thanks
 
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FAQ: Brightness Temperature, Thermal Spectrum

1. What is brightness temperature?

Brightness temperature is a measure of the amount of thermal radiation emitted by an object at a specific frequency. It is not the actual temperature of the object, but rather an equivalent temperature that represents the amount of radiation emitted.

2. How is brightness temperature related to thermal spectrum?

Brightness temperature is directly related to thermal spectrum, as it is a measure of the thermal radiation emitted by an object. The thermal spectrum is a graph that shows the amount of radiation emitted by an object at different frequencies, and the brightness temperature is the equivalent temperature that represents this amount of radiation.

3. What factors affect the brightness temperature of an object?

The brightness temperature of an object is affected by its temperature, emissivity (ability to emit radiation), and the frequency at which it is being observed. Additionally, the atmosphere and any surrounding objects can also influence the observed brightness temperature.

4. How is brightness temperature measured?

The brightness temperature of an object is typically measured using remote sensing techniques, such as satellite sensors or radiometers. These instruments detect the thermal radiation emitted by an object at different frequencies and use this information to calculate the corresponding brightness temperature.

5. What applications does brightness temperature have in science and technology?

Brightness temperature is used in a variety of scientific and technological applications, including weather forecasting, land surface temperature monitoring, and remote sensing of the Earth's surface. It is also used in astronomy to study the thermal properties of celestial objects and in materials science to measure the thermal properties of different materials.

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