Equations

The currently implemented equations are:

Gravitational-wave amplitude spin-down limit

This equation can be accessed from the equations() function using the name h0spindown.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("h0spindown", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s, distance=1.0 kpc)

Gravitational-wave amplitude spin-down limit.

Parameters

equation (str) – “h0spindown”

Keyword Arguments

• momentofinertia (float or Quantity) – Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: “izz”, “i38”.

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• rotationfdot (float or Quantity) – Source rotational frequency derivative. The default value is -1e-11 Hz / s. Alternative keyword names are: “frotdot”, “f1rot”, “f1spin”.

• distance (float or Quantity) – 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 equations() function using the name spindownluminosity.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("spindownluminosity", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s)

The spin-down luminosity of a pulsar.

Parameters

equation (str) – “spindownluminosity”

Keyword Arguments

• momentofinertia (float or Quantity) – Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: “izz”, “i38”.

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• rotationfdot (float or Quantity) – 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 equations() function using the name gwluminosity.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("gwluminosity", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, ellipticity=1e-06)

The gravitational-wave luminosity of a pulsar.

Parameters

equation (str) – “gwluminosity”

Keyword Arguments

• momentofinertia (float or Quantity) – Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: “izz”, “i38”.

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• ellipticity (float or Quantity) – 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 equations() function using the name characteristicage.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("characteristicage", brakingindex=3, rotationperiod=0.01 s, rotationpdot=1e-15)

The characteristic age of a pulsar.

Parameters

equation (str) – “characteristicage”

Keyword Arguments

• brakingindex (float or Quantity) – The braking index of a pulsar. The default value is 3. Alternative keyword names are: “n”.

• rotationperiod (float or Quantity) – Source rotational period. The default value is 0.01 s. Alternative keyword names are: “prot”, “p0rot”.

• rotationpdot (float or Quantity) – 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 equations() function using the name massquadrupole.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("massquadrupole", ellipticity=1e-06, momentofinertia=1e+38 kg m2)

Mass quadrupole.

Parameters

equation (str) – “massquadrupole”

Keyword Arguments

• ellipticity (float or Quantity) – Neutron star ellipticity. The default value is 1e-06. Alternative keyword names are: “ell”, “eps”, “epsilon”, “𝜀”.

• momentofinertia (float or Quantity) – 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 equations() function using the name ellipticityspindown.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("ellipticityspindown", momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, rotationfdot=-1e-11 Hz / s)

Spin-down limit for neutron star ellipticity.

Parameters

equation (str) – “ellipticityspindown”

Keyword Arguments

• momentofinertia (float or Quantity) – Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: “izz”, “i38”.

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• rotationfdot (float or Quantity) – 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 equations() function using the name brakingindex.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("brakingindex", rotationfrequency=50.0 Hz, rotationfddot=1e-23 Hz / s2, rotationfdot=-1e-11 Hz / s)

The braking index of a pulsar.

Parameters

equation (str) – “brakingindex”

Keyword Arguments

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 50.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• rotationfddot (float or Quantity) – Source rotational frequency second derivative. The default value is 1e-23 Hz / s2. Alternative keyword names are: “frotddot”, “f2rot”, “f2spin”.

• rotationfdot (float or Quantity) – 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 equations() function using the name h0.

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

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

The fiducial values defined for this equation are:

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

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 equations() can be used as

equations("h0", ellipticity=1e-06, momentofinertia=1e+38 kg m2, rotationfrequency=100.0 Hz, distance=1.0 kpc)

Gravitational wave amplitude.

Parameters

equation (str) – “h0”

Keyword Arguments

• ellipticity (float or Quantity) – Neutron star ellipticity. The default value is 1e-06. Alternative keyword names are: “ell”, “eps”, “epsilon”, “𝜀”.

• momentofinertia (float or Quantity) – Principal moment of inertia about the rotation axis. The default value is 1e+38 kg m2. Alternative keyword names are: “izz”, “i38”.

• rotationfrequency (float or Quantity) – Source rotational frequency. The default value is 100.0 Hz. Alternative keyword names are: “frot”, “spinfrequency”, “fspin”, “f0rot”, “f0spin”.

• distance (float or Quantity) – 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]

2(1,2,3)

Condon, J. J. and Ransom, S. M., 2016, Essential Radio Astronomy [ADS URL]

3

Ostriker, J. P. and Gunn, J. E., 1969, ApJ, 157, 1395 [ADS URL]

4

Owen. B. 2005, PhRvL, 95, 211101 [ADS URL]

5

Abbott, B. P., et al. 2019, ApJ, 879, 10 [ADS URL]

6

Jaranowski, P., Krolak, A., & Schutz, B. F. 1998, PhRvD, 58, 063001 [ADS URL]