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Phys3605 - Homework week 9

Spring 2026

Galactic dust consists of particles of 0.5 - 20 µm made of silicates and hydrocarbon molecules.

The dust is at a temperature T = 20 K and emits thermal radiation with this specific intensity

where Bν is the black body function. An experiment made several measurements of the specific intensity I at frequencies between 100 and 200 GHz. The data collected is stored in https://canvas.umn.edu/courses/519864/files/54660242?wrap=1. This data file should be linked in the homework assignment too, and is tabulated below in Table 1 (for reference only - you must write MATLAB code that reads the csv file).

Your goal is to determine the amplitude A and spectral index β and to indicate your level of certainty in this determination.

At frequencies between 100 and 200 GHz a black body at a temperature of 20 K is in the Rayleigh-Jeans part of the spectrum and the specific intensity is well approximated as

where k(= 1.38 × 10−23 J/K) is Boltzman’s constant and c(= 3 × 108 m/s) is the speed of light.

Submit a documented .mlx and .pdf file explaining how you found A and β using a linear fit and Matlab function(s) that you wrote to do the fitting. The following components must be included:

• An explanation of your approach.

• A Figure with the data and the linear fit.

• The parameters of the fit and their uncertainties.

• The χ 2 , reduced χ 2 , and at least a qualitative assessment of the goodness of fit.

• A Figure of the intensity data with the best fit spectrum given in Equation 1 over the relevant frequency range.

• All validation steps you have taken to ensure the analysis is correct.

• The results you would report for A and β and their uncertainties.

1Useful to remember: if

then

Similarly, let’s say

Obviously, if you use ln = loge instead of log10, everything is the same but ln e = 1.

Table 1: Measured values and uncertainties