代写天体物理-ASTR601代写-Astrophysics代写

ASTR601 – Astrophysics

Week 1 Tutorial: Planck’s Law

代写天体物理 Today this is called Rayleigh-Jeans’ law 1 and it works at wavelengths longer than that of visible light (e.g. radio).

In 1901, Max Planck (1858-1947) correctly predicted the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature, which can be expressed as brightness B as a function of wavelength λ and temperature T 代写天体物理

(1)

where

Run the provided Matlab code, which plots B (λ, T) over a range of wavelengths and temperatures. The plot should look like

Activities 代写天体物理

Adapt the provided Matlab code to verify Wien’s displacement law,

λmaxT = 2.897755 × 10−3 K m, (2)

by numerically estimating λmaxT for around fifive difffferent values of T spanning T = 100 K to T = 10, 000 K. Here, λmax is the wavelength at which B (λ, T) reaches its maximum amplitude.

Before Planck, Lord Rayleigh (1842-1919) arrived at a prediction of the shape of the black body spectrum

Today this is called Rayleigh-Jeans’ law 1 and it works at wavelengths longer than that of visible light (e.g. radio). Using the approximation e x ≈ 1 + x when x 1, derive Rayleigh-Jeans’ law from Planck’s law,equation 1.

3.Adapt the Matlab code to plot B (λ, T) on a log-log scale (logarithmic scale on both the x-axis and the y-axis), and expand the x-axis to include radio wavelengths up to 20 cm. Then plot BR−J (λ, T) for the same range of temperatures and wavelengths to verify that the Rayleigh-Jeans’approximation is accurate at long wavelengths. 代写天体物理

Extend the range of temperatures plotted to start at 1 K.
Plot the spectrum of the Cosmic Microwave Background (CMB).
At what wavelength does the CMB spectrum peak? 代写天体物理

Make a copy of the Matlab code and modify it to plot the black body spectrum as a function of frequency ν. To derive B (ν, T), note that the integral of the spectral radiance per unit wavelength over a range of wavelengths must be equal to the integral of the spectral radiance per unit frequency over the corresponding range of frequencies; i.e.

(3)

For the temperature of the CMB, fifind νmax (the frequency at which the CMB spectrum peaks) and convert this frequency to a wavelength, λ 0max.Compare λ 0max to λmax as predicted by Wien’s displacement law for the CMB temperature. Why are these two values difffferent?

1James Jeans (1877-1946) found and corrected a mistake in Rayleigh’s original derivation.

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