#########
Equations
#########

The currently implemented equations are:

Gravitational-wave amplitude spin-down limit
--------------------------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``h0spindown``.

The generated equation (Eqn. 5 in  [1]_) is:

.. math::

    h_0^{\rm sd} = \frac{\sqrt{10} \sqrt{G}}{2 c^{3/2}}\frac{I_{zz}^{1/2} \left|{\dot{f}_{\rm rot}}\right|^{1/2}}{d f_{\rm rot}^{1/2}}

The fiducial values defined for this equation are:

.. math::

    h_0^{\rm sd} = 2.55 \times 10^{-25} \; \mathrm{}\left(\frac{100 \; \mathrm{Hz}}{f_{\rm rot}}\right)^{1/2} \left(\frac{1 \; \mathrm{kpc}}{d}\right) \left(\frac{I_{zz}}{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}\right)^{1/2} \left(\frac{\left|{\dot{f}_{\rm rot}}\right|}{1 \times 10^{-11} \; \mathrm{\frac{Hz}{s}}}\right)^{1/2} 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("h0spindown", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s, distance=1.0 kpc)
    :noindex:
    
    Gravitational-wave amplitude spin-down limit.
    
    :param str equation: "h0spindown"
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfdot: Source rotational frequency derivative. The default value is -1e-11 Hz / s. Alternative keyword names are: "**frotdot**", "**f1rot**", "**f1spin**".
    :keyword float or ~astropy.units.quantity.Quantity distance: Distance to the source. The default value is 1.0 kpc. Alternative keyword names are: "**dist**", "**d**", "**r**".
    

The spin-down luminosity of a pulsar
------------------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``spindownluminosity``.

The generated equation (Eqn. 6.35 in  [2]_) is:

.. math::

    L_{\rm sd} = 4 \pi^{2}I_{zz} f_{\rm rot} \left|{\dot{f}_{\rm rot}}\right|

The fiducial values defined for this equation are:

.. math::

    L_{\rm sd} = 3.95 \times 10^{30} \; \mathrm{W}\left(\frac{I_{zz}}{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}\right) \left(\frac{f_{\rm rot}}{100 \; \mathrm{Hz}}\right) \left(\frac{\dot{f}_{\rm rot}}{-1 \times 10^{-11} \; \mathrm{\frac{Hz}{s}}}\right) 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("spindownluminosity", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s)
    :noindex:
    
    The spin-down luminosity of a pulsar.
    
    :param str equation: "spindownluminosity"
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfdot: Source rotational frequency derivative. The default value is -1e-11 Hz / s. Alternative keyword names are: "**frotdot**", "**f1rot**", "**f1spin**".
    

The gravitational-wave luminosity of a pulsar
---------------------------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``gwluminosity``.

The generated equation (Eqn. 7 in  [3]_) is:

.. math::

    L_{\rm gw} = \frac{2048 \pi^{6} G}{5 c^{5}}\varepsilon^{2} I_{zz}^{2} f_{\rm rot}^{6}

The fiducial values defined for this equation are:

.. math::

    L_{\rm gw} = 1.09 \times 10^{29} \; \mathrm{W}\left(\frac{I_{zz}}{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}\right)^{2} \left(\frac{f_{\rm rot}}{100 \; \mathrm{Hz}}\right)^{6} \left(\frac{\varepsilon}{1 \times 10^{-6} \; \mathrm{}}\right)^{2} 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("gwluminosity", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, ellipticity=1e-06)
    :noindex:
    
    The gravitational-wave luminosity of a pulsar.
    
    :param str equation: "gwluminosity"
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity ellipticity: Neutron star ellipticity. The default value is 1e-06. Alternative keyword names are: "**ell**", "**eps**", "**epsilon**", "**𝜀**".
    

The characteristic age of a pulsar
----------------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``characteristicage``.

The generated equation (Eqn. 6.31 in  [2]_) is:

.. math::

    \tau = \frac{P}{\dot{P} \left(n - 1\right)}

The fiducial values defined for this equation are:

.. math::

    \tau = 1.58 \times 10^{5} \; \mathrm{yr}\left(\frac{P}{0.01 \; \mathrm{s}}\right) \left(\frac{1 \times 10^{-15} \; \mathrm{}}{\dot{P}}\right) \left(\frac{2 \; \mathrm{}}{n - 1}\right) 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("characteristicage", brakingindex=3, rotationperiod=0.01 s, rotationpdot=1e-15)
    :noindex:
    
    The characteristic age of a pulsar.
    
    :param str equation: "characteristicage"
    :keyword float or ~astropy.units.quantity.Quantity brakingindex: The braking index of a pulsar. The default value is 3. Alternative keyword names are: "**n**".
    :keyword float or ~astropy.units.quantity.Quantity rotationperiod: Source rotational period. The default value is 0.01 s. Alternative keyword names are: "**prot**", "**p0rot**".
    :keyword float or ~astropy.units.quantity.Quantity rotationpdot: Source rotational period derivative. The default value is 1e-15. Alternative keyword names are: "**pdot**", "**p0dot**".
    

Mass quadrupole
---------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``massquadrupole``.

The generated equation (Eqn. 2 in  [4]_) is:

.. math::

    Q_{22} = \frac{\sqrt{30}}{4 \sqrt{\pi}}\varepsilon I_{zz}

The fiducial values defined for this equation are:

.. math::

    Q_{22} = 7.73 \times 10^{31} \; \mathrm{m^{2}\,kg}\left(\frac{I_{zz}}{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}\right) \left(\frac{\varepsilon}{1 \times 10^{-6} \; \mathrm{}}\right) 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("massquadrupole", ellipticity=1e-06, momentofinertia=1e+38 kg m2)
    :noindex:
    
    Mass quadrupole.
    
    :param str equation: "massquadrupole"
    :keyword float or ~astropy.units.quantity.Quantity ellipticity: Neutron star ellipticity. The default value is 1e-06. Alternative keyword names are: "**ell**", "**eps**", "**epsilon**", "**𝜀**".
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    

Spin-down limit for neutron star ellipticity
--------------------------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``ellipticityspindown``.

The generated equation (Eqn. A9 in  [5]_) is:

.. math::

    \varepsilon^{\rm sd} = \frac{\sqrt{10} c^{5/2}}{32 \pi^{2} \sqrt{G}}\frac{\left|{\dot{f}_{\rm rot}}\right|^{1/2}}{I_{zz}^{1/2} f_{\rm rot}^{5/2}}

The fiducial values defined for this equation are:

.. math::

    \varepsilon^{\rm sd} = 6.03 \times 10^{-6} \; \mathrm{}\left(\frac{\left|{\dot{f}_{\rm rot}}\right|}{1 \times 10^{-11} \; \mathrm{\frac{Hz}{s}}}\right)^{1/2} \left(\frac{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}{I_{zz}}\right)^{1/2} \left(\frac{100 \; \mathrm{Hz}}{f_{\rm rot}}\right)^{5/2} 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("ellipticityspindown", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s)
    :noindex:
    
    Spin-down limit for neutron star ellipticity.
    
    :param str equation: "ellipticityspindown"
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfdot: Source rotational frequency derivative. The default value is -1e-11 Hz / s. Alternative keyword names are: "**frotdot**", "**f1rot**", "**f1spin**".
    

The braking index of a pulsar
-----------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``brakingindex``.

The generated equation (Eqn. 6.35 in  [2]_) is:

.. math::

    n = \frac{\ddot{f}_{\rm rot} f_{\rm rot}}{\dot{f}_{\rm rot}^{2}}

The fiducial values defined for this equation are:

.. math::

    n = 5 \; \mathrm{}\left(\frac{f_{\rm rot}}{50 \; \mathrm{Hz}}\right) \left(\frac{-1 \times 10^{-11} \; \mathrm{\frac{Hz}{s}}}{\dot{f}_{\rm rot}}\right)^{2} \left(\frac{\ddot{f}_{\rm rot}}{1 \times 10^{-23} \; \mathrm{\frac{Hz}{s^{2}}}}\right) 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("brakingindex", rotationfrequency=50.0 Hz, rotationfddot=1e-23 Hz / s2, rotationfdot=-1e-11 Hz / s)
    :noindex:
    
    The braking index of a pulsar.
    
    :param str equation: "brakingindex"
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 50.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfddot: Source rotational frequency second derivative. The default value is 1e-23 Hz / s2. Alternative keyword names are: "**frotddot**", "**f2rot**", "**f2spin**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfdot: Source rotational frequency derivative. The default value is -1e-11 Hz / s. Alternative keyword names are: "**frotdot**", "**f1rot**", "**f1spin**".
    

Gravitational wave amplitude
----------------------------

This equation can be accessed from the :func:`~cweqgen.equations.equations` function using
the name ``h0``.

The generated equation (Eqn. 23 in  [6]_) is:

.. math::

    h_0 = \frac{16 G}{c^{4}} \pi^{2}\frac{\varepsilon I_{zz}}{d} f_{\rm rot}^{2}

The fiducial values defined for this equation are:

.. math::

    h_0 = 4.23 \times 10^{-26} \; \mathrm{}\left(\frac{1 \; \mathrm{kpc}}{d}\right) \left(\frac{I_{zz}}{1 \times 10^{38} \; \mathrm{m^{2}\,kg}}\right) \left(\frac{f_{\rm rot}}{100 \; \mathrm{Hz}}\right)^{2} \left(\frac{\varepsilon}{1 \times 10^{-6} \; \mathrm{}}\right) 

.. note::

    These fiducial values are just those defined within this package and may not be representative
    of fiducial values used elsewhere in the literature.

To generate the equation as calculated at particular values, the
:func:`~cweqgen.equations.equations` can be used as

.. py:function:: equations("h0", ellipticity=1e-06, momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, distance=1.0 kpc)
    :noindex:
    
    Gravitational wave amplitude.
    
    :param str equation: "h0"
    :keyword float or ~astropy.units.quantity.Quantity ellipticity: Neutron star ellipticity. The default value is 1e-06. Alternative keyword names are: "**ell**", "**eps**", "**epsilon**", "**𝜀**".
    :keyword float or ~astropy.units.quantity.Quantity momentofinertia: Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: "**izz**", "**i38**".
    :keyword float or ~astropy.units.quantity.Quantity rotationfrequency: Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: "**frot**", "**spinfrequency**", "**fspin**", "**f0rot**", "**f0spin**".
    :keyword float or ~astropy.units.quantity.Quantity distance: Distance to the source. The default value is 1.0 kpc. Alternative keyword names are: "**dist**", "**d**", "**r**".
    

References
----------

.. [1] Aasi, A., et al. 2014, ApJ, 785, 119 [`ADS URL <https://ui.adsabs.harvard.edu/abs/2014ApJ...785..119A/abstract>`__]

.. [2] Condon, J. J. and Ransom, S. M., 2016, Essential Radio Astronomy [`ADS URL <https://ui.adsabs.harvard.edu/abs/2016era..book.....C/abstract>`__]

.. [3] Ostriker, J. P. and Gunn, J. E., 1969, ApJ, 157, 1395 [`ADS URL <https://ui.adsabs.harvard.edu/abs/1969ApJ...157.1395O/abstract>`__]

.. [4] Owen. B. 2005, PhRvL, 95, 211101 [`ADS URL <https://ui.adsabs.harvard.edu/abs/2005PhRvL..95u1101O/abstract>`__]

.. [5] Abbott, B. P., et al. 2019, ApJ, 879, 10 [`ADS URL <https://ui.adsabs.harvard.edu/abs/2019ApJ...879...10A/abstract>`__]

.. [6] Jaranowski, P., Krolak, A., & Schutz, B. F. 1998, PhRvD, 58, 063001 [`ADS URL <https://ui.adsabs.harvard.edu/abs/1998PhRvD..58f3001J/abstract>`__]