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I have an assignment, where I'm supposed to estimate the temperature of a star in at least two different ways, and determine which of the approaches gives the most reliable estimate. All that is given is the star's brightness at some random wavelengths. One way of estimating would be finding the peak brightness from the data, and use the corresponding wavelength as peak wavelength and solving

$$T = \frac{b}{\lambda_{max}}$$

So I've got 20 different wavelengths, ranging from 90nm to 1000nm. How would one go one about estimating T from a blackbody spectrum? Should I graph one with the given wavelengths and try different values for T until it fits the data?

However, I can't think of any other approaches with so little data.

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    $\begingroup$ Further information/detail required. $\endgroup$
    – ProfRob
    Mar 11, 2020 at 11:59
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    $\begingroup$ How many wavelengths? What are they? Can you fit a blackbody spectrum to them? $\endgroup$
    – uhoh
    Mar 11, 2020 at 12:05
  • $\begingroup$ So I've got 20 different wavelengths, ranging from 90nm to 1000nm. How would one go one about estimating T from a blackbody spectrum? Should I graph one with the given wavelengths and try different values for T until it fits the data? $\endgroup$
    – amateur
    Mar 11, 2020 at 13:42
  • $\begingroup$ If you got values for 20 wavelength, make a fit for a black body spectrum through those values and you're done. It might also be interesting to know whether the "random" wavelength are not so random and correspond to typical wavelengths of typical filters, U, B, V for example. $\endgroup$
    – planetmaker
    Mar 11, 2020 at 14:58
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    $\begingroup$ Please add the additional information in your comments to the question itself (edit the question). If people have to read through comments to find important information it reduces the likelihood your question will be answered. $\endgroup$
    – StephenG - Help Ukraine
    Mar 11, 2020 at 17:16

1 Answer 1

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  1. Use Wien's displacement law - as you suggest.

  2. Let's assume that the spectrum you have been given incorporates almost all the flux from the star. This might be ok, so long as the flux is heading towards small numbers at each end of the spectrum? If so, then you can use the temperature-independent shape of a blackbody function to argue that some fixed fraction of the flux is contained below a wavelength which depends only on the temperature. This is a more robust version of Wien's law.

For example 50% of the flux should lie below a wavelength of $4107/T$ $\mu$m.

Thus you sum up your 20 points, using Simpson's rule or Trapezium rule if they are unevenly spaced or have gaps. Then you work out how much flux there is up to the wavelength of each of the wavelength points in turn. When you reach the one where 50% of the total flux is below it, then the temperature is $4107/\lambda$, where $\lambda$ is in $\mu$m.

In principle method 2. is more robust, because you are always trying to find the effective temperature - which is the temperature of a blackbody that gives the same observed integrated flux. Since method 2 is an integral method, it is less sensitive to the detailed shape of the spectrum. Which is good, because stars are not blackbodies.

In contrast, method 1 is really at the mercy of how close to a blackbody the spectrum is.

A possible problem with method 2. though, is that your spectrum does not contain all of the flux. In which case your temperature will be an overestimate (assuming the missing flux is longward of 1000 nm).

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