r"""Supernova SED models from the Villar et al. (2019) parametric light curve."""
from typing import ClassVar
from astropy import units as u
from uvex_transients.models._typing import CGSParameterValue, FloatArray
from uvex_transients.models._util_functions import cooling_temperature_cgs
from uvex_transients.models.core.base import SpectralModel
from uvex_transients.models.core.parameters import Parameter
from uvex_transients.models.core.priors import LogNormalPrior
from uvex_transients.models.lightcurves.generic import VillarLightcurve
from uvex_transients.models.spectra.thermal import BlackbodySpectrum
__all__ = [
"VillarCoolingBlackbodySED",
]
[docs]
class VillarCoolingBlackbodySED(SpectralModel):
r"""
Supernova light curve with an evolving blackbody photosphere.
This model combines the phenomenological bolometric light curve of
:class:`~uvex_transients.models.lightcurves.generic.VillarLightcurve` (the
parametric form introduced by :footcite:p:`2019ApJ...884...83V`) with a
time-dependent blackbody spectral shape,
.. math::
L_\nu(\nu, t)
=
L_\mathrm{bol}(t)\,
S_\mathrm{BB}\!\left[\nu, T(t)\right],
where :math:`S_\mathrm{BB}` is the normalized blackbody spectrum provided
by :class:`~uvex_transients.models.spectra.thermal.BlackbodySpectrum`.
The photospheric temperature evolves according to
.. math::
T(t)
=
T_\mathrm{floor}
+
\left(T_0-T_\mathrm{floor}\right)
\left(1+\frac{t}{\tau_T}\right)^{-\alpha_T}.
Thus, the temperature begins at :math:`T_0` at :math:`t=0` and approaches
:math:`T_\mathrm{floor}` asymptotically at late times. The parameters
:math:`\tau_T` and :math:`\alpha_T` control the characteristic cooling
timescale and the rate of the decline, respectively.
Because the blackbody shape is normalized independently of luminosity,
the bolometric and spectral components remain exactly separable:
.. math::
\int L_\nu(\nu,t)\,d\nu = L_\mathrm{bol}(t).
Consequently, :meth:`_eval_bolometric` delegates directly to
:class:`~uvex_transients.models.lightcurves.generic.VillarLightcurve`, while
:meth:`_eval_spectrum` evaluates
:class:`~uvex_transients.models.spectra.thermal.BlackbodySpectrum` at the
time-dependent temperature :math:`T(t)`. No numerical frequency
integration is required to recover the bolometric luminosity.
The model is intended as a generic phenomenological description of
supernova-like transients whose continuum can be approximated by a cooling
photosphere. It does not attempt to model photospheric-radius evolution,
line blanketing, recombination physics, nebular emission, or detailed
radiative transfer.
.. rubric:: Parameters
The model parameters are summarized below.
.. list-table::
:header-rows: 1
:widths: 18 18 64
* - Parameter
- Symbol
- Description
* - ``amplitude``
- :math:`A`
- Overall luminosity normalization.
* - ``t0``
- :math:`t_0`
- Reference time at which the logistic rise is centered.
* - ``gamma``
- :math:`\gamma`
- Duration of the plateau, measured from t0.
* - ``beta``
- :math:`\beta`
- Linear slope of the plateau.
* - ``tau_rise``
- :math:`\tau_\mathrm{rise}`
- Logistic rise timescale.
* - ``tau_fall``
- :math:`\tau_\mathrm{fall}`
- Exponential decline timescale, after the plateau.
* - ``T0``
- :math:`T_0`
- Photospheric temperature at t=0 (the T(t) -> T0 limit, not
literally T at peak).
* - ``T_floor``
- :math:`T_\mathrm{floor}`
- Asymptotic late-time photospheric temperature (T(t) -> T_floor
as t -> infinity).
* - ``tau_T``
- :math:`\tau_T`
- Photospheric cooling timescale.
* - ``alpha_T``
- :math:`\alpha_T`
- Photospheric cooling power-law index.
Notes
-----
The default parameter distributions are broad, phenomenological starting
points rather than subtype-specific empirical priors.
Subclasses may define only the entries of :attr:`_DEFAULT_PARAMETERS` they
wish to modify. Their parameter definitions are merged with the complete
parameter set of ``VillarCoolingBlackbodySED`` during subclass creation.
This makes it straightforward to define supernova-subtype variants by
changing only the parameters whose distributions differ between
populations.
The temperature law assumes :math:`T_0 > T_\mathrm{floor}` for a cooling
photosphere, although this ordering is not enforced by the model itself.
References
----------
.. footbibliography::
"""
_DEFAULT_PARAMETERS: ClassVar[dict[str, Parameter]] = {
**VillarLightcurve._DEFAULT_PARAMETERS,
"T0": Parameter(
prior=LogNormalPrior(mean=0.0, sigma=0.3),
scale=1.2e4 * u.K,
description="Photospheric temperature at t=0 (the T(t) -> T0 limit, not literally T at peak).",
latex=r"T_0",
),
"T_floor": Parameter(
prior=LogNormalPrior(mean=0.0, sigma=0.3),
scale=6e3 * u.K,
description="Asymptotic late-time photospheric temperature (T(t) -> T_floor as t -> infinity).",
latex=r"T_\mathrm{floor}",
),
"tau_T": Parameter(
prior=LogNormalPrior(mean=0.0, sigma=0.5),
scale=15.0 * u.day,
description="Photospheric cooling timescale.",
latex=r"\tau_T",
),
"alpha_T": Parameter(
prior=LogNormalPrior(mean=0.0, sigma=0.3),
scale=1.5,
description="Photospheric cooling power-law index.",
latex=r"\alpha_T",
),
}
# -------------------------------------- #
# Subclass Merging #
# -------------------------------------- #
def __init_subclass__(cls, **kwargs) -> None:
"""
Merge a subtype subclass's own `_DEFAULT_PARAMETERS` on top of this class's full parameter set.
Parameters
----------
**kwargs
Forwarded to :meth:`object.__init_subclass__` unchanged; this
class declares no class-keyword-argument options of its own.
"""
super().__init_subclass__(**kwargs)
own = cls.__dict__.get("_DEFAULT_PARAMETERS")
if own is not None:
cls._DEFAULT_PARAMETERS = {
**VillarCoolingBlackbodySED._DEFAULT_PARAMETERS,
**own,
}
# -------------------------------------- #
# Cooling Law: T(t) #
# -------------------------------------- #
@classmethod
def _temperature_cgs(
cls,
t: FloatArray,
*,
T0: CGSParameterValue,
T_floor: CGSParameterValue,
tau_T: CGSParameterValue,
alpha_T: CGSParameterValue,
**_ignored: CGSParameterValue,
) -> FloatArray:
r"""
:math:`T(t) = T_\mathrm{floor} + (T_0 - T_\mathrm{floor})(1 + t/\tau_T)^{-\alpha_T}`.
Parameters
----------
t : numpy.ndarray
Time since explosion, in seconds.
T0, T_floor, tau_T, alpha_T : float or numpy.ndarray
This model's parameter values, in cgs units; see the class docstring.
**_ignored
Any other model parameter values, ignored.
Returns
-------
numpy.ndarray
:math:`T(t)`, in Kelvin.
"""
return cooling_temperature_cgs(t, T0=T0, T_floor=T_floor, timescale=tau_T, alpha=alpha_T)
# -------------------------------------- #
# Bolometric Luminosity: L_bol(t) #
# -------------------------------------- #
@classmethod
def _eval_bolometric(cls, t: FloatArray, **parameters: CGSParameterValue) -> FloatArray:
r"""
:math:`\log L_\mathrm{bol}(t)`, delegated directly to :class:`VillarLightcurve`.
Exact -- no integration needed.
Parameters
----------
t : numpy.ndarray
Time since explosion, in seconds.
**parameters
This model's parameter values, in cgs units.
Returns
-------
numpy.ndarray
The natural log of :math:`L_\mathrm{bol}(t)`, in erg/s.
"""
lightcurve_parameters = {name: parameters[name] for name in VillarLightcurve._DEFAULT_PARAMETERS}
return VillarLightcurve._eval(t, **lightcurve_parameters)
# -------------------------------------- #
# Normalized Spectral Shape: S(nu, t) #
# -------------------------------------- #
@classmethod
def _eval_spectrum(cls, nu: FloatArray, t: FloatArray, **parameters: CGSParameterValue) -> FloatArray:
r"""
:math:`\log S(\nu, T(t))`, delegated to :class:`BlackbodySpectrum`.
Evaluated at this ``t``'s own cooling-law temperature.
Parameters
----------
nu : numpy.ndarray
Frequency, in Hz.
t : numpy.ndarray
Time since explosion, in seconds.
**parameters
This model's parameter values, in cgs units.
Returns
-------
numpy.ndarray
The natural log of the normalized spectral shape, in 1/Hz.
"""
temperature = cls._temperature_cgs(t, **parameters)
return BlackbodySpectrum._eval(nu, temperature=temperature)
# -------------------------------------- #
# Spectral Luminosity: L_nu(nu, t) #
# -------------------------------------- #
@classmethod
def _eval(cls, nu: FloatArray, t: FloatArray, **parameters: CGSParameterValue) -> FloatArray:
r"""
:math:`\log L_\nu(\nu, t) = \log L_\mathrm{bol}(t) + \log S(\nu, T(t))`.
Parameters
----------
nu : numpy.ndarray
Frequency, in Hz.
t : numpy.ndarray
Time since explosion, in seconds.
**parameters
This model's parameter values, in cgs units.
Returns
-------
numpy.ndarray
The natural log of :math:`L_\nu(\nu, t)`, in erg/s/Hz.
"""
return cls._eval_bolometric(t, **parameters) + cls._eval_spectrum(nu, t, **parameters)
