lazygun37
commented
6 years ago Hi guys,
just time for a brief response, but hopefully I can cover the key bits:
We are having trouble creating this Lorentzian spectrum to start with, we initially tried to create it from the Probability density function (PDF) from the equation for the Cauchy distribution found here: https://en.wikipedia.org/wiki/Cauchy_distribution
With x0 being the local viscous frequency and the full width half maximum Q being (gamma = Q/2). We had trouble trying to modify the astroML Timmer and Koenig function
https://github.com/astroML/astroML/blob/master/astroML/time_series/generate.py
As the power law of 1/omega^beta is not well explicitly defined.
One thought we had was that perhaps line 49 was the square root of the power law and thus we could simply replace it with the square root of the Lorentzian distribution.
- Yes, the Lorentzian is basically the same thing as the Cauchy distribution. See the wikipedia page under "Characterization", where it says "In physics, a three-parameter Lorentzian function is often used". It's just that the Cauchy distribution is defined as a "proper" probability distribution (which integrates to unity), whereas the Lorentzian is allowed to have a variable height (and hence variable integral), as defined by I.
- And yes, I would also try initially mainly changing line 49 in the astroML code; instead of doing 1/omega*(0.5beta) there, you want to call your (square root of the) Lorentzian function
- As you seem to suspect, the reason for the "0.5" in the power law is that what's happening in the code is that you define the *Fourier Transform", not the power spectrum, with the latter being the square of the former. So indeed you want the square root of the Lorentzian.
We also were wondering about what occurs what the height of the Lorentzians would be as well as what exactly occurs when two or more of the curves overlap.
- Regarding the normalization of the Lorentzians, see Section 2.2 of Arevalo & Uttley, starting at "The normalization of these signal PSDs...." -- The bottom line is really that they adopt a normalization that ends up giving them a constant standard deviation in the light curve for each annulus. They achieve this by appropriate normalizing the PDF (it's possible to normalize the PSD so it's integral is RMS). But you could also probably just do it by not worrying about the normalizing much and just normalizing the resulting light curve.
- Regarding overlap: basically, don't worry about it. Obviously, in the limit of infinite numbers of annuli, the overlap is always guaranteed, i.e. you get a smooth PSD. More generally, you'll get a smooth PSD whenever you have a combination of enough annuli and broad enough Lorentzians. But, for now, I'm not especially concerned about that. Once you have your code set up, we can easily test this numerically.
Hope that helps!
Cheers,
Christian
Currently, for every radius, we generate m_dot (the local small mass accretion rate) from the Timmer and Koenig method based on a PSD derived from a power law. This time series has no dependence on any of the properties of the disc and is random throughout all time.
Instead, we wish to produce the m_dot variations based on a spectrum linked to the properties of the disc. This in theory can be done by using a Lorentzian based spectrum where each radius has a Lorentzian peak associated that is related to its viscous frequency.
Causing a light curve to be produced where most of the variability is based around the local viscous frequency (timescale).
We are having trouble creating this Lorentzian spectrum to start with, we initially tried to create it from the Probability density function (PDF) from the equation for the Cauchy distribution found here: https://en.wikipedia.org/wiki/Cauchy_distribution
With x0 being the local viscous frequency and the full width half maximum Q being (gamma = Q/2). We had trouble trying to modify the astroML Timmer and Koenig function
https://github.com/astroML/astroML/blob/master/astroML/time_series/generate.py
As the power law of 1/omega^beta is not well explicitly defined.
One thought we had was that perhaps line 49 was the square root of the power law and thus we could simply replace it with the square root of the Lorentzian distribution.
We also were wondering about what occurs what the height of the Lorentzians would be as well as what exactly occurs when two or more of the curves overlap.