TeXify meaure_eccentricity tref_in/fref_in

gw_eccentricity/gw_eccentricity.py

@@ -66,51 +66,63 @@ def measure_eccentricity(tref_in=None,
  extra_kwargs=None):
  """Measure eccentricity and mean anomaly from a gravitational waveform.
 
- Eccentricity is measured using the GW frequency omega22(t) =
- dphi22(t)/dt, where phi22(t) is the phase of the (2, 2) waveform
- mode. We currently only allow time-domain, nonprecessing waveforms. We
- evaluate omega22(t) at pericenter times, t_pericenters, and build a
- spline interpolant omega22_pericenters(t) using those data
- points. Similarly, we build omega22_apocenters(t) using omega22(t) at
- the apocenter times, t_apocenters.
-
- Using omega22_pericenters(t) and omega22_apocenters(t), we first
- compute e_omega22(t), as described in Eq.(4) of arXiv:2302.11257. We
- then use e_omega22(t) to compute the eccentricity egw(t) using Eq.(8)
- of arXiv:2302.11257. Mean anomaly is defined using t_pericenters, as
- described in Eq.(10) of arXiv:2302.11257.
-
- To find t_pericenters/t_apocenters, one can look for extrema in different
- waveform data, like omega22(t) or Amp22(t), the amplitude of the (2, 2)
- mode. Pericenters correspond to the local maxima, while apocenters
- correspond to the local minima in the data. The method option
+ Eccentricity is measured using the GW frequency :math:`\\omega_{22}(t) =
+ \\frac{d\\phi_{22}(t)}{dt}`, where :math:`\\phi_{22}(t)` is the phase of
+ the :math:`(2, 2)` waveform mode. We currently only allow time-domain,
+ nonprecessing waveforms. We evaluate :math:`\\omega_{22}(t)` at pericenter
+ times, :math:`t^{\\text{pericenters}}`, and build a spline interpolant
+ :math:`\\omega_{22}^{\\text{pericenters}}(t)` using those data
+ points. Similarly, we build :math:`\\omega_{22}^{\\text{apocenters}}(t)`
+ using :math:`\\omega_{22}(t)` at the apocenter times,
+ :math:`t^{\\text{apocenters}}`.
+
+ Using :math:`\\omega_{22}^{\\text{pericenters}}(t)` and
+ :math:`\\omega_{22}^{\\text{apocenters}}(t)`, we first compute
+ :math:`e_{\\omega_{22}}(t)`, as described in Eq.(4) of
+ `arXiv:2302.11257`_. We then use :math:`e_{\\omega_{22}}(t)` to compute the
+ eccentricity :math:`e_{\\text{gw}}` using Eq.(8) of
+ `arXiv:2302.11257`_. Mean anomaly :math:`l_{\\text{gw}}` is defined using
+ :math:`t^{\\text{pericenters}}`, as described in Eq.(10) of
+ `arXiv:2302.11257`_.
+
+ To find :math:`t^{\\text{pericenters}}`/:math:`t^{\\text{apocenters}}`, one
+ can look for extrema in different waveform data, like
+ :math:`\\omega_{22}(t)` or :math:`A_{22}(t)`, the amplitude of the
+ :math:`(2, 2)` mode. Pericenters correspond to the local maxima, while
+ apocenters correspond to the local minima in the data. The method option
  (described below) lets the user pick which waveform data to use to find
- t_pericenters/t_apocenters.
+ :math:`t^{\\text{pericenters}}`/:math:`t^{\\text{apocenters}}`.
+
+ .. _arXiv:2302.11257: https://arxiv.org/abs/2302.11257
 
  Parameters
  ----------
- tref_in
+ tref_in : float or array_like
  Input reference time at which to measure eccentricity and mean anomaly.
- Can be a single float or an array.
 
- fref_in
+ fref_in : float or array_like
  Input reference GW frequency at which to measure the eccentricity and
- mean anomaly. Can be a single float or an array. Only one of
- tref_in/fref_in should be provided.
+ mean anomaly. Only one of `tref_in`/`fref_in` should be provided.
+
+ Given an `fref_in`, we find the corresponding `tref_in` such that
+
+ .. math::
+
+ \\omega_{22}^{\\text{average}}(t^{\\text{ref}}_{\\text{in}})
+ = 2 \\pi f^{\\text{ref}}_{\\text{in}}
 
- Given an fref_in, we find the corresponding tref_in such that
- omega22_average(tref_in) = 2 * pi * fref_in. Here, omega22_average(t)
- is a monotonically increasing average frequency obtained from the
- instantaneous omega22(t). omega22_average(t) defaults to the orbit
- averaged omega22, but other options are available (see
- omega22_averaging_method below).
+ Here, :math:`\\omega_{22}^{\\text{average}}(t)` is a monotonically
+ increasing average frequency obtained from the instantaneous
+ :math:`\\omega_{22}(t)`. :math:`\\omega_{22}^{\\text{average}}(t)`
+ defaults to the orbit averaged :math:`\\omega_{22}(t)`, but other
+ options are available (see `omega22_averaging_method` below).
 
  Eccentricity and mean anomaly measurements are returned on a subset of
- tref_in/fref_in, called tref_out/fref_out, which are described below.
- If dataDict is provided in dimensionless units, tref_in should be in
- units of M and fref_in should be in units of cycles/M. If dataDict is
- provided in MKS units, t_ref should be in seconds and fref_in should be
- in Hz.
+ `tref_in`/`fref_in`, called `tref_out`/`fref_out`, which are described
+ below. If `dataDict` is provided in dimensionless units, `tref_in`
+ should be in units of :math:`M` and `fref_in` should be in units of
+ cycles/:math:`M`. If `dataDict` is provided in MKS units, `t_ref`
+ should be in seconds and `fref_in` should be in Hz.
 
  method : str, default: ``Amplitude``
  Which waveform data to use for finding extrema. Options are: