- #1
jcap
- 170
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I am trying to understand how the width of a supernova light curve depends on the redshift of its component frequencies.
Let us make the simple assumption that the light curve is Gaussian. The inverse Fourier transform of a Gaussian is given by:
$$\large e^{-\alpha t^2}=\int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ift}\ df$$
Now if all the components of the light curve are redshifted by a factor [itex]k[/itex] then I think the right-hand side of the above equation becomes:
$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ikft}\ df$$
I now change variables in the integral using:
$$f'=kf$$
The above integral becomes the inverse Fourier transform of a modified Gaussian curve:
$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha k^2}}e^{-\frac{(\pi f')^2}{\alpha k^2}}e^{2\pi if't}\ df'$$
Thus it seems that if the components are redshifted by a factor [itex]k[/itex] the light curve transforms in the following way:
$$\large e^{-\alpha t^2} \rightarrow e^{-\alpha k^2t^2}$$
Is this correct?
Let us make the simple assumption that the light curve is Gaussian. The inverse Fourier transform of a Gaussian is given by:
$$\large e^{-\alpha t^2}=\int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ift}\ df$$
Now if all the components of the light curve are redshifted by a factor [itex]k[/itex] then I think the right-hand side of the above equation becomes:
$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha}}e^{-\frac{(\pi f)^2}{\alpha}}e^{2\pi ikft}\ df$$
I now change variables in the integral using:
$$f'=kf$$
The above integral becomes the inverse Fourier transform of a modified Gaussian curve:
$$\large \int_{-\infty}^{\infty}\sqrt{\frac{\pi}{\alpha k^2}}e^{-\frac{(\pi f')^2}{\alpha k^2}}e^{2\pi if't}\ df'$$
Thus it seems that if the components are redshifted by a factor [itex]k[/itex] the light curve transforms in the following way:
$$\large e^{-\alpha t^2} \rightarrow e^{-\alpha k^2t^2}$$
Is this correct?
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