diff --git a/README.md b/README.md index e140b4b9..8f8f398f 100644 --- a/README.md +++ b/README.md @@ -21,10 +21,15 @@ You can install via pip using Or you can clone or download the repository and then use `python setup.py install` or `pip install .` -Documentation +Help ===== -Some basic documentation can be found [here](https://pulsar-spectra.readthedocs.io/en/latest/) +The documentation can be found [here](https://pulsar-spectra.readthedocs.io/en/latest/) +Credit +===== +If you use pulsar_spectra for your research please give credit by citing [Swainston et al 2012, PASA, 39, e056](https://ui.adsabs.harvard.edu/abs/2022arXiv220913324S/abstract) and the [publications of the data](https://pulsar-spectra.readthedocs.io/en/latest/catalogue.html#papers-included-in-our-catalogue) used in your spectral fits. + +Until there is a more appropriate method for crediting software development and maintainance, please also consider including me as a co-author on publications which rely on pulsar_spectra. Catalogue data ===== @@ -34,8 +39,3 @@ Instead, you should search through the literature to find all papers that contai the pulsar and confirm all of those papers are in the catalogue. You can find a list of the papers in the catalogue [here](https://pulsar-spectra.readthedocs.io/en/latest/catalogue.html#papers-included-in-our-catalgoue) If you would like to add a new paper to the catalogue read [the guide](https://pulsar-spectra.readthedocs.io/en/latest/catalogue.html#adding-papers) - - -Status -===== -[![Documentation Status](https://readthedocs.org/projects/pulsar-spectra/badge/?version=latest)](https://pulsar-spectra.readthedocs.io/en/latest/?badge=latest) \ No newline at end of file diff --git a/docs/bandwidth_intergration.rst b/docs/bandwidth_intergration.rst index dd626026..99bc3ea6 100644 --- a/docs/bandwidth_intergration.rst +++ b/docs/bandwidth_intergration.rst @@ -5,7 +5,7 @@ Pulsar spectral fitting often assumes that the reported average flux densities a the flux density at one specific (usually central) frequency, whereas in reality, they are averaged over some finite bandwidth. This assumption becomes increasingly inaccurate for wider fractional bandwidths. For this reason we have expanded the catalogue's database to include the bandwidth of all detections and -expanded our equations to model the integrated flux across the band. +expanded our equations to model the integrated flux density across the band. When the bandwidth integration is used --------------------------------------- @@ -40,18 +40,19 @@ You can see how the bandwidth of each flux density measurement is now clearly di Derivations ----------- -If \alpha measurement is reported along with \alpha bandwidth, then the correct way to fit models is to find the expected mean flux across the band for each model, +If a flux density measurement is reported along with a bandwidth, then the correct way to fit models is to find the expected mean flux density across the band for each model, .. math:: - S_{avg} = \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{min}} S_v\,\text{d}\nu, + S_\rm{avg} = \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{min}} S_\nu\,\text{d}\nu, where :math:`\rm{BW} = \nu_\text{min} - \nu_\text{min}`. -The evaluation of this expression for each model follows. +The evaluation of this expression for each of the models currently implemented in pulsar_spectra follows. +Derivations for the log-parabolic are also included, although this model is deactivated in pulsar_spectra by default. How to use sympy to help with derivations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -`Sympy `_ is an excellent tool for performing differentation and simple intergration like so: +`Sympy `_ is an excellent tool for performing differentation and simple intergration. An example is given below: .. code:: @@ -59,26 +60,32 @@ How to use sympy to help with derivations f = c * (v/v0)^a * exp( \alpha / beta * (v/vpeak)^(-beta) ) f2 = f.diff(v).diff(v).simplify() -Which will output the second differentatial: +This will output the second derivative of the specified function: .. code:: a*c*(v/v0)^a*(v/vpeak)^(-2*(a + (v/vpeak)^(2*(a - 1) + (v/vpeak)^beta*(-2*a + beta + 1))*exp(a*(v/vpeak)^(-beta)/beta)/v^2 -Intergration Derivations ------------------------- - -.. _simple_power_law_integrate: +Integration of the model functions +---------------------------------- +In the following subsections, bandwidth integrations are performed directly using the model functions which are detailed in Swainston et al. (2022). Simple power law ~~~~~~~~~~~~~~~~ +The simple power law model is + +.. math:: + + S_\nu = c \left( \frac{\nu}{\nu_0} \right)^\alpha, + +where :math:`\nu_0` is the reference frequency, :math:`\alpha` is the spectral index, and :math:`c` is a constant. +The bandwidth integration for this model is relatively simple, as shown below: .. math:: - S_\nu &= c \left( \frac{\nu}{\nu_0} \right)^\alpha, \\ S_\text{avg} &= \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} c \left( \frac{\nu}{\nu_0} \right)^\alpha \,\text{d}\nu, \\ &= \frac{\nu_0}{\rm{BW}} \left[\frac{c}{\alpha+1} \left(\frac{\nu}{\nu_0}\right)^{\alpha + 1} \right]_{\nu_\text{min}}^{\nu_\text{max}} \\ - &= \frac{\nu_0}{\rm{BW}} \frac{c}{\alpha+1} \left( \left(\frac{{\nu_\text{max}}}{\nu_0}\right)^{\alpha + 1} - \left(\frac{{\nu_\text{min}}}{\nu_0}\right)^{\alpha + 1} \right) \\ + &= \frac{\nu_0}{\rm{BW}} \frac{c}{\alpha+1} \left[ \left(\frac{{\nu_\text{max}}}{\nu_0}\right)^{\alpha + 1} - \left(\frac{{\nu_\text{min}}}{\nu_0}\right)^{\alpha + 1} \right] \\ &= \frac{c({\nu_\text{max}}^{\alpha+1} - {\nu_\text{min}}^{\alpha+1})}{\rm{BW}\,\nu_0^\alpha(\alpha+1)}. .. _broken_power_law_intergral: @@ -86,6 +93,9 @@ Simple power law Broken power law ~~~~~~~~~~~~~~~~ +The broken power law model is two connected power laws with spectral indices :math:`\nu_1` and :math:`nu_2` +and a break frequency :math:`\nu_b`, + .. math:: S_\nu &= c\begin{cases} @@ -93,56 +103,59 @@ Broken power law \left( \frac{\nu}{\nu_0} \right)^{\alpha_2} \left( \frac{{\nu_b}}{\nu_0} \right)^{\alpha_1-\alpha_2} & \mathrm{otherwise} \\ \end{cases}. - -If :math:`{\nu_\text{min}} < {\nu_\text{max}} \le{\nu_b}`, then :math:`S_\text{avg}` is identical to the simple power law with the substitution :math:`\alpha \leftarrow \alpha_1`: +If :math:`{\nu_\text{min}} < {\nu_\text{max}} \le{\nu_b}`, then :math:`S_\text{avg}` is identical to the simple power law with the substitution :math:`\alpha \leftarrow \alpha_1` : .. math:: S_\text{avg} = \frac{c({\nu_\text{max}}^{\alpha_1+1} - {\nu_\text{min}}^{\alpha_1+1})}{\rm{BW}\,\nu_0^{\alpha_1}(\alpha_1+1)}. -If both :math:`{\nu_b} \le {\nu_\text{min}} < {\nu_\text{max}}`, then +If both :math:`{\nu_b} \le {\nu_\text{min}} < {\nu_\text{max}}`, then the subtitution :math:`\alpha \leftarrow \alpha_2` is performed, +and an additional factor is introduced to match the power law with the break frequency: .. math:: S_\text{avg} = \frac{c({\nu_\text{max}}^{\alpha_2+1} - {\nu_\text{min}}^{\alpha_2+1})}{\rm{BW}\,\nu_0^{\alpha_2}(\alpha_2+1)} \left( \frac{{\nu_b}}{\nu_0} \right)^{\alpha_1-\alpha_2}. - -In the final case, when :math:`{\nu_\text{min}} < {\nu_b} < {\nu_\text{max}}`, +In the final case, when :math:`{\nu_\text{min}} < {\nu_b} < {\nu_\text{max}}`, the solution is a combination of the above: .. math:: - S_\text{avg} = \frac{c({\nu_b}^{\alpha_1+1} - {\nu_\text{min}}^{\alpha_1+1})}{({\nu_b} - {\nu_\text{min}})\,\nu_0^{\alpha_1}(\alpha_1+1)} + \frac{c({\nu_\text{max}}^{\alpha_2+1} - {\nu_b}^{\alpha_2+1})}{({\nu_\text{max}} - {\nu_b})\,\nu_0^{\alpha_2}(\alpha_2+1)} \left( \frac{{\nu_b}}{\nu_0} \right)^{\alpha_1-\alpha_2}. + S_\text{avg} = \frac{c({\nu_b}^{\alpha_1+1} - {\nu_\text{min}}^{\alpha_1+1})}{({\nu_b} - {\nu_\text{min}})\,\nu_0^{\alpha_1}(\alpha_1+1)} + \frac{c({\nu_\text{max}}^{\alpha_2+1} - {\nu_b}^{\alpha_2+1})}{({\nu_\text{max}} - {\nu_b})\,\nu_0^{\alpha_2}(\alpha_2+1)} \left( \frac{{\nu_b}}{\nu_0} \right)^{\alpha_1-\alpha_2}, +where the factors :math:`(\nu_b-\nu_\rm{min})^{-1}` and :math:`(\nu_\rm{max}-\nu_b)^{-1}` replace the :math:`\rm{BW}^{-1}` in normalising the integrated flux density. Log-parabolic spectrum ~~~~~~~~~~~~~~~~~~~~~~ +The log-parabolic spectrum is a parabola in log-space, with the form: .. math:: \log_{10} S_\nu - &= \alpha \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + - b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c \\ + = a \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c, + +where :math:`a` is the curvature parameter, :math:`b` is the spectral index for :math:`a=0`, and :math:`c` is a constant. +This model can be re-expressed to be linear in :math:`S_\nu` as + +.. math:: + S_\nu &= 10^{a \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c} \\ &= e^{\ln 10 \left(a \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c\right)} \\ &= Ce^{\ln 10 \left(a \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right )\right)}, - -where :math:`C = e^{c\ln 10} = 10^c`. +where :math:`C = e^{c\ln 10} = 10^c`. The :math:`\log_{10}` terms can be expressed in terms of the natural logarithm as .. math:: S_\nu &= Ce^{\ln 10 \left(a \left [ \frac{\ln\left ( \frac{\nu}{\nu_0} \right )}{\ln 10} \right]^2 + b \, \frac{\ln \left ( \frac{\nu}{\nu_0} \right )}{\ln 10}\right)} \\ &= Ce^{\left(\frac{a}{\ln 10} \left [ \ln\left ( \frac{\nu}{\nu_0} \right )\right]^2 + b \, \ln \left ( \frac{\nu}{\nu_0} \right )\right)}. - -In this form, the integration becomes \emph{slightly} easier (at least, WolframAlpha gives an answer!): +In this form, the integration becomes slightly easier. WolframAlpha then returns the generic solution .. math:: \int e^{A(\ln x)^2 + B\ln x}\,\text{d}x = \frac{\sqrt{\pi} e^{-\frac{(B+1)^2}{4A}} \text{erfi} \left(\frac{2A\ln x + B + 1}{2\sqrt{A}}\right)}{2\sqrt{A}}. - In our case, this works out to .. math:: @@ -155,16 +168,21 @@ In our case, this works out to High-frequency cut-off power law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +This model is a power law spectrum with a high-frequency cut-off occurring at the cut-off frequency :math:`\nu_c` : + +.. math:: + + S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \left ( 1 - \frac{\nu}{\nu_c} \right ),\qquad \nu < \nu_c. + +The bandwidth integration is performed as follows: .. math:: - S_\nu &= c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \left ( 1 - \frac{\nu}{\nu_c} \right ),\qquad \nu < \nu_c, \\ S_\text{avg} &= \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \left ( 1 - \frac{\nu}{\nu_c} \right ) \,\text{d}\nu \\ &= -\frac{c}{\rm{BW} \nu_0^\alpha} \left[ \frac{\nu^{\alpha + 1}}{\alpha + 1} + \frac{\nu^{\alpha + 2}}{\nu_c (\alpha + 2)}\right]_{\nu_\text{min}}^{\nu_\text{max}} \\ - &= -\frac{c}{\rm{BW} \nu_0^\alpha} \left( \frac{{\nu_\text{max}}^{\alpha + 1} - {\nu_\text{min}}^{\alpha + 1}}{\alpha + 1} + \frac{{\nu_\text{max}}^{\alpha + 2} - {\nu_\text{min}}^{\alpha + 2}}{\nu_c (\alpha + 2)}\right ) \\ - + &= -\frac{c}{\rm{BW} \nu_0^\alpha} \left( \frac{{\nu_\text{max}}^{\alpha + 1} - {\nu_\text{min}}^{\alpha + 1}}{\alpha + 1} + \frac{{\nu_\text{max}}^{\alpha + 2} - {\nu_\text{min}}^{\alpha + 2}}{\nu_c (\alpha + 2)}\right ). -sympy solution: +sympy provides the solution: .. code:: @@ -172,81 +190,86 @@ sympy solution: .. math:: - S_\text{avg} &= \left( \frac{c \nu}{\rm{BW}\nu_c} \right) \left ( \frac{\nu}{\nu_0} \right)^ \alpha \left ( \frac{- \alpha \nu + \alpha \nu_c - \nu + 2 \nu_c}{ (\alpha + 1)(\alpha + 2)} \right)\\ + S_\text{avg} = \left( \frac{c \nu}{\rm{BW}\nu_c} \right) \left ( \frac{\nu}{\nu_0} \right)^ \alpha \left ( \frac{- \alpha \nu + \alpha \nu_c - \nu + 2 \nu_c}{ (\alpha + 1)(\alpha + 2)} \right). .. _low_frequency_turn_over_power_law_intergral: Low-frequency turn-over power law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +The low-frequency turn-over introduces an exponential cut-off to the power law model at low frequencies, of the form .. math:: - S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta} \right ]. + S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ], - -Again with \alpha little help from WolframAlpha, defining +where :math:`\alpha` is the spectral index, :math:`\beta` is a parameter describing the smoothness of the turn-over, and :math:`\nu_\rm{peak}` is the turn-over frequency. +To perform this integration, we define the following parameters: .. math:: X &= \left( \frac{\nu}{\nu_0} \right)^{\alpha}, \\ - Y &= -\frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta}, \\ - Z &= -\frac{\alpha + 1}{\beta}, + Y &= -\frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta}, \\ + Z &= -\frac{\alpha + 1}{\beta}. -we have +WolframAlpha returns the solution .. math:: S_\text{avg} &= \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} cX e^{-Y} \,\text{d}\nu \\ &= \frac{c}{\rm{BW}}\left[\frac{\nu X Y^{-Z}}{\beta} \Gamma(Z, Y) \right]_{\nu_\text{min}}^{\nu_\text{max}}, -where :math:`\Gamma(a,x)`` is the incomplete gamma function. +where :math:`\Gamma(a,x)`` is the upper incomplete gamma function. .. _double_turn_over_spectrum_intergral: Double turn-over spectrum ~~~~~~~~~~~~~~~~~~~~~~~~~ - +The double turn-over spectrum combines the low-frequency turn-over and high-frequency cut-off into a single model. +It takes the following form: .. math:: - S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] \left ( 1 - \frac{\nu}{\nu_c} \right ) ,\qquad \nu < \nu_c, - + S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ] \left ( 1 - \frac{\nu}{\nu_c} \right ) ,\qquad \nu < \nu_c. +We first re-arrange as follows: .. math:: S_\text{avg} - &= \frac{c}{\rm{BW}}\int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] \left ( 1 - \frac{\nu}{\nu_c} \right )\,\text{d}\nu \\ - &= - \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \,\text{d}\nu \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] \,\text{d}\nu - - \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \,\text{d}\nu \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] \frac{\nu}{\nu_c} \,\text{d}\nu \\ - &= - \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} Xe^{-Y} \,\text{d}\nu - - \frac{c\nu_0}{\rm{BW}\,\nu_c} \int_{\nu_\text{min}}^{\nu_\text{max}} X^\prime e^{-Y} \,\text{d}\nu \\ + &= \frac{c}{\rm{BW}}\int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ] \left ( 1 - \frac{\nu}{\nu_c} \right )\,\text{d}\nu \\ &= - \frac{c}{\rm{BW}}\left[\frac{\nu X Y^{-Z}}{\beta} \Gamma(Z, Y) \right]_{\nu_\text{min}}^{\nu_\text{max}} - - \frac{c\nu_0}{\rm{BW}\,\nu_c}\left[\frac{\nu X^\prime Y^{-Z^\prime}}{\beta} \Gamma(Z^\prime, Y) \right]_{\nu_\text{min}}^{\nu_\text{max}}, + \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ] \,\text{d}\nu -{} \\ + &\qquad \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ] \frac{\nu}{\nu_c} \,\text{d}\nu \\ -where +This integral can then be written in terms of the paramters .. math:: X &= \left( \frac{\nu}{\nu_0} \right)^{\alpha}, & - Y &= -\frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta}, & + Y &= -\frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta}, & Z &= -\frac{\alpha + 1}{\beta}, \\ X^\prime &= \left( \frac{\nu}{\nu_0} \right)^{\alpha+1}, & & & - Z^\prime &= -\frac{\alpha + 2}{\beta}, + Z^\prime &= -\frac{\alpha + 2}{\beta}. +WolframAlpha returns the solution -Taylor Expansion Derivations ----------------------------- +.. math:: + S_\text{avg} + &= \frac{c}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} Xe^{-Y} \,\text{d}\nu - + \frac{c\nu_0}{\rm{BW}\,\nu_c} \int_{\nu_\text{min}}^{\nu_\text{max}} X^\prime e^{-Y} \,\text{d}\nu \\ + &= \frac{c}{\rm{BW}}\left[\frac{\nu X Y^{-Z}}{\beta} \Gamma(Z, Y) \right]_{\nu_\text{min}}^{\nu_\text{max}} - + \frac{c\nu_0}{\rm{BW}\,\nu_c}\left[\frac{\nu X^\prime Y^{-Z^\prime}}{\beta} \Gamma(Z^\prime, Y) \right]_{\nu_\text{min}}^{\nu_\text{max}}. + +Derivation of the Taylor-expanded model functions +------------------------------------------------- Some of the above integrals involve functions that may be tricky to implement in practice. The following Taylor expansions allow for easier implementation, at the cost of accuracy for wideband measurements. Here, we derive Taylor expansions about an arbitrary "centre" frequency, :math:`{\nu_\text{ctr}}` : - S_\nu \approx S_{\nu_{ctr}} + S_{\nu_{ctr}}^\prime(\nu - {\nu_\text{ctr}}) + \frac{1}{2} S_{\nu_{ctr}}^{\prime\prime}(\nu - {\nu_\text{ctr}})^2 + \frac{1}{6} S_{\nu_{ctr}}^{\prime\prime\prime}(\nu - {\nu_\text{ctr}})^3 + \cdots +.. math:: + S_\nu \approx S_{\nu_{ctr}} + S_{\nu_{ctr}}^\prime(\nu - {\nu_\text{ctr}}) + \frac{1}{2} S_{\nu_{ctr}}^{\prime\prime}(\nu - {\nu_\text{ctr}})^2 + \frac{1}{6} S_{\nu_{ctr}}^{\prime\prime\prime}(\nu - {\nu_\text{ctr}})^3 + \cdots, where :math:`S_{\nu_{ctr}}^{(n)} = S^{(n)}({\nu_\text{ctr}})` is shorthand for the :math:`n` th derivative of :math:`S_\nu` with respect to frequency, evaluated at :math:`{\nu_\text{ctr}}` . @@ -256,20 +279,19 @@ In general, the bandwidth integral will then be S_\text{avg} &\approx \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} S_\nu\,\text{d}\nu \\ - &\approx \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left( + &\approx \frac{1}{\rm{BW}} \int_{\nu_\text{min}}^{\nu_\text{max}} \left[ S_{\nu_{ctr}} + S_{\nu_{ctr}}^\prime(\nu - {\nu_\text{ctr}}) + \frac{1}{2} S_{\nu_{ctr}}^{\prime\prime}(\nu - {\nu_\text{ctr}})^2 + \frac{1}{6} S_{\nu_{ctr}}^{\prime\prime\prime}(\nu - {\nu_\text{ctr}})^3 + \cdots - \right)\,\text{d}\nu \\ + \right]\,\text{d}\nu \\ &\approx \frac{1}{\rm{BW}} \left[ S_{\nu_{ctr}}\nu + \frac{S_{\nu_{ctr}}^\prime}{2}(\nu - {\nu_\text{ctr}})^2 + \frac{S_{\nu_{ctr}}^{\prime\prime}}{3}(\nu - {\nu_\text{ctr}})^3 + \frac{S_{\nu_{ctr}}^{\prime\prime\prime}}{4}(\nu - {\nu_\text{ctr}})^4 + \cdots \right]_{\nu_\text{min}}^{\nu_\text{max}} \\ - &\approx \frac{1}{\rm{BW}} \left( + &\approx \frac{1}{\rm{BW}} \left[ 2S_{\nu_{ctr}}\left(\frac{\rm{BW}}{2}\right) + \frac{2S_{\nu_{ctr}}^{\prime\prime}}{3}\left(\frac{\rm{BW}}{2}\right)^3 + \cdots - \right) \\ + \right] \\ &= S_{\nu_{ctr}} + \frac{S_{\nu_{ctr}}^{\prime\prime}}{3}\left(\frac{\rm{BW}}{2}\right)^2 + \cdots - We see that every other term cancels (due to the symmetry of the integrand), and the final sum is therefore .. math:: @@ -277,38 +299,49 @@ We see that every other term cancels (due to the symmetry of the integrand), and S_\text{avg} = \sum_{k=0}^\infty \frac{S_{\nu_{ctr}}^{(2k)}}{2k+1}\left(\frac{\rm{BW}}{2}\right)^{2k}. -This formula can then be simply implemented for each model by computing its ``even'' derivatives. +This formula can then be simply implemented for each model by computing its "even" derivatives. This is done for each model in the following subsections. -[To-do: Calculate the residual error for \alpha given truncation, for each of the models. Also need to consider the radius of convergence (esp. for models that are defined with cut-off frequencies).] +[To-do: Calculate the residual error for a given truncation, for each of the models. Also need to consider the radius of convergence (esp. for models that are defined with cut-off frequencies).] Simple power law ~~~~~~~~~~~~~~~~ +The derivatives of the simple power-law model, + +.. math:: + + S_\nu = c \left( \frac{\nu}{\nu_0} \right)^\alpha, + +take the following form: .. math:: - S_\nu &= c \left( \frac{\nu}{\nu_0} \right)^\alpha \\ S_\nu^\prime &= \alpha c \frac{\nu^{\alpha - 1}}{\nu_0^\alpha} - = \frac{\alpha S_\nu}{\nu} \\ + = \frac{\alpha S_\nu}{\nu}, \\ S_\nu^{\prime\prime} &= \alpha(\alpha - 1) c \frac{\nu^{\alpha - 2}}{\nu_0^\alpha} - = \frac{\alpha(\alpha - 1)S_\nu}{\nu^2} \\ + = \frac{\alpha(\alpha - 1)S_\nu}{\nu^2}, \\ &\vdots \notag \\ S_\nu^{(k)} - &= \frac{\alpha!}{(\alpha - k)!}\frac{S_\nu}{\nu^k} - - + &= \frac{\alpha!}{(\alpha - k)!}\frac{S_\nu}{\nu^k}. Broken power law ~~~~~~~~~~~~~~~~ -This one is too awkward to do using \alpha Taylor expansion, I reckon. +This derivation was not performed due to the complexity of performing this Taylor expansion. Log-parabolic spectrum ~~~~~~~~~~~~~~~~~~~~~~ -For brevity, I will use the shorthands +The log-parabolic spectrum is + +.. math:: + + \log_{10} S_\nu = a\left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + + b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c. + +For brevity, the following shorthands will be used: .. math:: @@ -316,7 +349,7 @@ For brevity, I will use the shorthands Y &\equiv \frac{2a}{\ln 10}. -Note that +Also note that the derivatives of these parameters are .. math:: @@ -324,18 +357,14 @@ Note that \qquad\text{and}\qquad Y^\prime = 0. - -The first four derivatives are: +The first four derivatives of the model are: .. math:: - \log_{10} S_\nu - &= \alpha \left [ \log_{10} \left ( \frac{\nu}{\nu_0} \right ) \right]^2 + - b \, \log_{10} \left ( \frac{\nu}{\nu_0} \right ) + c \\ \frac{S_\nu^\prime}{S_\nu\ln10} &= \left(2a\log_{10} \left ( \frac{\nu}{\nu_0} \right ) + b\right) \left( \frac{1}{\nu\ln 10}\right) - = \frac{X}{\nu\ln 10} \\ + = \frac{X}{\nu\ln 10}, \\ S_\nu^\prime &= \frac{S_\nu X}{\nu} \\ S_\nu^{\prime\prime} @@ -343,17 +372,17 @@ The first four derivatives are: \frac{S_\nu^\prime X}{\nu} - \frac{S_\nu X}{\nu^2} + \frac{S_\nu X^\prime}{\nu} \\ - &= \frac{S_\nu}{\nu^2}\left( X^2 - X + Y \right) \\ + &= \frac{S_\nu}{\nu^2}\left( X^2 - X + Y \right), \\ S_\nu^{\prime\prime\prime} &= \frac{S_\nu^\prime}{\nu^2}\left( X^2 - X + Y \right) - \frac{2S_\nu}{\nu^3}\left( X^2 - X + Y \right) + \frac{S_\nu}{\nu^2}\left( 2XX^\prime - X^\prime \right) \\ - &= \frac{S_\nu}{\nu^3}\left( X^3 - 3X^2 + 3XY + 2X - 3Y \right) \\ + &= \frac{S_\nu}{\nu^3}\left( X^3 - 3X^2 + 3XY + 2X - 3Y \right), \\ S_\nu^{\prime\prime\prime\prime} &= \frac{S_\nu^\prime}{\nu^3}\left( X^3 - 3X^2 + 3XY + 2X - 3Y \right) -{} \\ &\qquad\frac{3S_\nu}{\nu^4}\left( X^3 - 3X^2 + 3XY + 2X - 3Y \right) +{} \\ &\qquad\frac{S_\nu}{\nu^3}\left( 3X^2X^\prime - 6XX^\prime + 3X^\prime Y + 2X^\prime \right) \\ - &= \frac{S_\nu}{\nu^4}\left( X^4 - 6X^3 + 6X^2 Y + 11X^2 - 18XY - 6X + 11Y + 3Y^2 \right) + &= \frac{S_\nu}{\nu^4}\left( X^4 - 6X^3 + 6X^2 Y + 11X^2 - 18XY - 6X + 11Y + 3Y^2 \right). .. _high_frequency_cut_off_power_law_taylor: @@ -361,27 +390,7 @@ The first four derivatives are: High-frequency cut-off power law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -This one is really just the sum of two simple power laws: - -.. math:: - - S_\nu - &= c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \left ( 1 - \frac{\nu}{\nu_c} \right ), \\ - &= c\left( \frac{\nu}{\nu_0} \right)^{\alpha} - \frac{c\nu_0}{\nu_c}\left( \frac{\nu}{\nu_0} \right)^{\alpha + 1}. - - -The derivatives are: - -.. math:: - - S_\nu^{(k)} - = \frac{c}{\nu_0^k} \frac{\alpha!}{(\alpha - k)!} - \left(\frac{\nu}{\nu_0}\right)^{\alpha - k}\left(1 - \frac{\nu}{\nu_c}\right) - - \frac{kc}{\nu_0^{k-1}\nu_c} \frac{\alpha!}{(\alpha - k + 1)!} - \left(\frac{\nu}{\nu_0}\right)^{\alpha - k + 1} - - -A new attempt +This high-frequency cut-off model can be rewritten as .. math:: @@ -389,112 +398,61 @@ A new attempt &= c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \left ( 1 - \frac{\nu}{\nu_c} \right ), \\ &= \left( \frac{c}{\nu_0^{\alpha}} \right ) \left (\nu^{\alpha} - \frac{\nu^{\alpha + 1}}{\nu_c} \right). -Deratives we need are: +The first three even derivatives are then: .. math:: S_\nu^{\prime\prime} &= \left( \frac{c \alpha }{\nu_0^{\alpha}} \right ) - \left( + \left[ (\alpha - 1) \nu^{\alpha -2} - \frac{(\alpha+1) \nu^{\alpha -1}}{\nu_c} - \right)\\ + \right],\\ S_\nu^{\prime\prime\prime\prime} - &= \left( \frac{c \alpha (\alpha - 1) (\alpha - 2) }{\nu_0^{\alpha}} \right ) - \left( + &= \left[ \frac{c \alpha (\alpha - 1) (\alpha - 2) }{\nu_0^{\alpha}} \right ] + \left[ (\alpha - 3) \nu^{\alpha - 4} - \frac{(\alpha+1) \nu^{\alpha -3}}{\nu_c} - \right) \\ + \right],\\ S_\nu^{\prime\prime\prime\prime\prime\prime} - &= \left( \frac{c + &= \left[ \frac{c \alpha (\alpha - 1) (\alpha - 2) (\alpha - 3) (\alpha - 4) } - {\nu_0^{\alpha}} \right ) - \left( + {\nu_0^{\alpha}} \right] + \left[ (\alpha - 5) \nu^{\alpha - 6} - \frac{(\alpha+1) \nu^{\alpha -5}}{\nu_c} - \right) + \right]. .. _low_frequency_turn_over_power_law_taylor: Low-frequency turn-over power law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Shorthands: - -.. math:: - - X &= \left( \frac{\nu}{\nu_c} \right)^{-\beta} & - Y &= 1 - X \\ - X^\prime - &= -\frac{\beta}{\nu_c} \left( \frac{\nu}{\nu_c} \right)^{-\beta - 1} - = -\frac{\beta X}{\nu} & - Y^\prime - &= -X^\prime - = \frac{\beta X}{\nu} - - -Derivatives: - +The low-frequency turn-over model is .. math:: - S_\nu^\prime - &= \frac{c\alpha}{\nu_0} \left( \frac{\nu}{\nu_0} \right)^{\alpha - 1} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta} \right ] + - c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta} \right ] \left(-\frac{\alpha}{\nu_c} \left( \frac{\nu}{\nu_c} \right)^{-\beta - 1} \right) \\ - &= \frac{\alpha S_\nu}{\nu} - \frac{\alpha S_\nu}{\nu_c} \left( \frac{\nu}{\nu_c} \right)^{-\beta - 1} \\ - &= \frac{\alpha S_\nu}{\nu}\left( 1 - \left( \frac{\nu}{\nu_c} \right)^{-\beta} \right) - = \frac{\alpha S_\nu}{\nu}\left( 1 - X \right) - = \frac{\alpha S_\nu Y}{\nu} - + S_\nu = c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ]. - -.. math:: - - S_\nu^{\prime\prime} - &= \frac{\alpha S_\nu^\prime Y}{\nu} - - \frac{\alpha S_\nu Y}{\nu^2} + - \frac{\alpha S_\nu Y^\prime}{\nu} \\ - &= \frac{\alpha^2 S_\nu Y^2}{\nu^2} - - \frac{\alpha S_\nu Y}{\nu^2} + - \frac{\alpha \beta S_\nu X}{\nu^2} \\ - &= \frac{\alpha S_\nu}{\nu^2} \left [ \alpha Y^2 - Y + \beta X \right ] - - - -.. math:: - - S_\nu^{\prime\prime\prime} - &= - \frac{\alpha S_\nu^\prime}{\nu^2}\left [ \alpha Y^2 - Y + \beta X \right ] - - \frac{2\alpha S_\nu}{\nu^3}\left [ \alpha Y^2 - Y + \beta X \right ] + \frac{\alpha S_\nu}{\nu^2}\left [ 2\alpha Y Y^\prime - Y^\prime + \beta X^\prime \right ] \\ - &= - \frac{\alpha S_\nu}{\nu^3}\alpha Y \left [ \alpha Y^2 - Y + \beta X \right ] - - \frac{\alpha S_\nu}{\nu^3}2\left [ \alpha Y^2 - Y + \beta X \right ] + \frac{\alpha S_\nu}{\nu^3}\left [ 2\alpha Y - 1 - \beta \right ] \beta X \\ - &= - \frac{\alpha S_\nu}{\nu^3}\bigg( \alpha^2 Y^3 - 3\alpha Y^2 + (3\alpha\beta X + 2)Y - \beta X(3 + \beta) - \bigg) - - -Shorthands: +To take the derivatives, we introduce the following shorthand: .. math:: X = \left( \frac{\nu}{\nu_{peak}} \right)^{\beta} +The first three even derivatives are then: .. math:: - S_\nu &= - c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_c} \right)^{-\beta} \right ].\\ S_\nu^{\prime\prime} &= \left(\frac{\alpha c}{\nu^2}\right) - \left (\frac{\nu}{v0} \right)^\alpha - \left(\frac{\nu}{\nu_{peak}} \right)^{-2 \beta} + \left (\frac{\nu}{\nu_0} \right)^\alpha + \left(\frac{\nu}{\nu_\rm{peak}} \right)^{-2 \beta} \left[\alpha + - \left(\frac{\nu}{\nu_{peak}} \right)^{2*\beta} (\alpha - 1) + - \left(\frac{\nu}{\nu_{peak}} \right)^{\beta} (-2\alpha + \beta + + \left(\frac{\nu}{\nu_\rm{peak}} \right)^{2\beta} (\alpha - 1) + + \left(\frac{\nu}{\nu_\rm{peak}} \right)^{\beta} (-2\alpha + \beta + 1)\right] - \exp\left[\left(\frac{\alpha}{\beta} \right) \left(\frac{\nu}{\nu_{peak}} \right)^{-\beta}\right]\\ + \exp\left[\left(\frac{\alpha}{\beta} \right) \left(\frac{\nu}{\nu_\rm{peak}} \right)^{-\beta}\right]\\ &= S_\nu \left(\frac{\alpha}{\nu^2}\right) X^{-2} \left[\alpha + X^{2} (\alpha - 1) + X (-2\alpha + \beta + 1)\right]\\ S_\nu^{\prime\prime\prime\prime} &= @@ -622,29 +580,34 @@ Shorthands: + 15 ) + \alpha^5 - \bigg ] + \bigg ]. .. _double_turn_over_spectrum_taylor: Double turn over spectrum ~~~~~~~~~~~~~~~~~~~~~~~~~ -Shorthands: +The double turn-over spectrum model is .. math:: + S_\nu = + c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ] + \left ( 1 - \frac{\nu}{\nu_c} \right ). - X &= \left( \frac{\nu}{\nu_{peak}} \right)^{\beta} \\ - Y &= (\nu -\nu_c)\\ - Z &= c\left(\frac{\nu}{\nu_0}\right)^\alpha \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] +For this derivation, the shorthands are: + +.. math:: + X &= \left( \frac{\nu}{\nu_\rm{peak}} \right)^{\beta}, \\ + Y &= (\nu -\nu_c),\\ + Z &= c\left(\frac{\nu}{\nu_0}\right)^\alpha \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_\rm{peak}} \right)^{-\beta} \right ]. +The first two even derivatives are then: .. math:: - S_\nu &= - c\left( \frac{\nu}{\nu_0} \right)^{\alpha} \exp\left [ \frac{\alpha}{\beta} \left( \frac{\nu}{\nu_{peak}} \right)^{-\beta} \right ] \left ( 1 - \frac{\nu}{\nu_c} \right )\\ S_\nu^{\prime\prime} - &= Z \frac{\alpha}{\nu^2\nu_c X^2} (-\alpha Y - 2\nu X^2 + 2\nu X + X^2(1 - \alpha) Y + X Y(2\alpha - \beta - 1))\\ + &= Z \frac{\alpha}{\nu^2\nu_c X^2} (-\alpha Y - 2\nu X^2 + 2\nu X + X^2(1 - \alpha) Y + X Y(2\alpha - \beta - 1)),\\ S_\nu^{\prime\prime\prime\prime} &= Z \frac{\alpha}{X^4\nu^4\nu_c} @@ -727,5 +690,5 @@ Shorthands: &\dots \alpha^3 \nu_c - \alpha^3 \nu - \bigg ] + \bigg ].