pimf.initialmassfunction

`pimf.initialmassfunction`

Some python classes to analytically do Initial Mass Function integrals, and easily keep track of the type of IMF.

Module Contents

Classes

`InitialMassFunction`
`PowerLawIMF`	Implement a power law IMF of the form $\xi(m)dm = \xi_0 m^\alpha dm$.
`ChabrierIMF`	Implement a Chabrier IMF of the form \[\begin{split} \xi(m)dm = \xi_0 \begin{cases} \frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right), & m \leq 1M_\odot\\ \xi_\textrm{continuity} m^{\alpha}, & 1M_\odot \leq m \end{cases} \end{split}\]
`BrokenPowerLawIMF`	Implement a broken power law IMF of the form: \[\begin{split} \xi(m)dm = \xi_0\begin{cases} m^{\alpha_1}, & m \leq M_\textrm{transition} \\ \xi_\textrm{continuity}m^{\alpha_2}, & M_\textrm{transition} \leq m \end{cases} \end{split}\]
`L3IMF`	Implement an L3 IMF of the form \[ \xi(m)dm = \xi_0 \left(\frac{m}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{m}{\mu}\right)^{1-\alpha}\right)^{-\beta} dm, \] introduced in Maschberger (2013).
`LognormalIMF`	Implement a Lognormal IMF of the form \[ \xi(m)dm = \xi_0\frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right) \]
`GeneralisedGammaIMF`	Implement a Generalised Gamma Function IMF of the form $\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]$.
`ExponentialCutoffPowerLawIMF`	Implement a Generalised Gamma Function IMF of the form $\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]$.

Functions

`popII_IMFs`
`popIII_IMFs`
`main`

Data

__all__

API

pimf.initialmassfunction.__all__ = ['InitialMassFunction', 'PowerLawIMF', 'ChabrierIMF', 'BrokenPowerLawIMF', 'L3IMF', 'LognormalIMF', ...

class pimf.initialmassfunction.InitialMassFunction

abstractmethod __call__()

__str__(): Return str(self).

abstractmethod integrate(Mmin, Mmax)

abstractmethod integrate_product(Mmin, Mmax)

_check_Mmin_Mmax(Mmin, Mmax)

set_normalisation(ξ0): Directly set the normalisation constant $\xi_0$, storing the flag normalised.

normalise_by_number(Mmin, Mmax, value=1): Normalise the IMF by number, so $\int^{M_\mathrm{max}}_{M_\mathrm{min}} \xi(m)\mathrm{d}m = x$, where x is a user provided value, default 1.

normalise_by_mass(Mmin, Mmax, value=1): Normalise the IMF by mass, so $\int^{M_\mathrm{max}}_{M_\mathrm{min}} m\xi(m)\mathrm{d}m = x$, where x is a user provided value, default 1.

mean_mass(Mmin, Mmax): Calculate mean mass of an IMF.

\[M_\mathrm{mean} = \frac{\int^{M_\mathrm{max}}_{M_\mathrm{min}} m\xi(m)\mathrm{d}m}{\int^{M_\mathrm{max}}_{M_\mathrm{min}} m\xi(m)\mathrm{d}m}\]

Parameters

Mmin : int or float Minimum mass. Mmax : int or float. Maximum mass.

Returns

float The mean mass of a star for the IMF.

integrate_linear_piecewise_interpolated_product(a, b, x, y, extrapolate_grid=False)

Integrate between a and b, assuming the IMF is multiplied by a linear piecewise function defined by the points (xi, yi).

Assume f s.t. f(xi) = yi, where f is linear and piecewise. This function solves the integral $\int^b_a f(m)\xi(m)dm$.

Passing extrapolate_grid=True (False by default) will use the slope between (x0, y0), (x1, y1) for x < x0, and similarly for x > xN.

abstractmethod inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

_plot_helper(**kwargs): SHOULD NOT BE CONSIDERED PART OF PUBLIC API

class pimf.initialmassfunction.PowerLawIMF(alpha=-2.35, normalisation='number', normalisation_value=1, Mmin=0.1, Mmax=100)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement a power law IMF of the form $\xi(m)dm = \xi_0 m^\alpha dm$.

You can specify if you want to normalise by mass or number, and the range. If a number is passed to normalisation then it is used If no normalisation is specified, then we use “number”, which with normalisation_value=1 will yield a PDF.

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

With the default slope of -2.35 this represents a Salpeter (1955) IMF.

Initialization

__str__(): Return str(self).

__call__(M): Calculate the value $\xi(m) = \xi_0 m^\alpha$.

integrate(Mmin, Mmax): Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} \xi(m)dm = \xi_0 \frac{M_{max}^{\alpha+1} - M_{min}^{\alpha+1}}{\alpha + 1}$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax): Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} m\xi(m)dm = \xi_0 \frac{M_{max}^{\alpha+2} - M_{min}^{\alpha+2}}{\alpha + 2}$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

_integrate_general(Mmin, Mmax, n)

_integrate_log(Mmin, Mmax)

_inverse_cdf_log(x): Special case of inverse_cdf for alpha = -1.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.ChabrierIMF(mc=0.079, sigma=0.69, alpha=-2.3, normalisation='number', normalisation_value=1, Mmin=0.1, Mmax=100)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement a Chabrier IMF of the form

\[\begin{split} \xi(m)dm = \xi_0 \begin{cases} \frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right), & m \leq 1M_\odot\\ \xi_\textrm{continuity} m^{\alpha}, & 1M_\odot \leq m \end{cases} \end{split}\]

You can specify if you want to normalise by mass or number, and the range. If a number is passed to normalisation then it is used If no normalisation is specified, then we take $\xi_0 = 1$.

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

With the default values this represents a Chabrier (2003) IMF.

Initialization

__str__(): Return str(self).

__call__(M): Calculate the value $\xi(m)$.

integrate(Mmin, Mmax): Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} \xi(m)dm = FILL IN$.

Note

The above equation doesn’t take into account that Mmin and Mmax could both be above/below the transition mass, but the function call does.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax): Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} m\xi(m)dm = FILL IN$.

Note

The above equation doesn’t take into account that Mmin and Mmax could both be above/below the transition mass, but the function call does.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

calculate_continuity_constant(): Calculate the continuity constant $\xi_\textrm{continuity}$ such that the lognormal and power law sections are continuous.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.BrokenPowerLawIMF(alpha1=-1.3, alpha2=-2.3, Mtransition=0.5, normalisation='number', normalisation_value=1, Mmin=0.1, Mmax=100)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement a broken power law IMF of the form:

\[\begin{split} \xi(m)dm = \xi_0\begin{cases} m^{\alpha_1}, & m \leq M_\textrm{transition} \\ \xi_\textrm{continuity}m^{\alpha_2}, & M_\textrm{transition} \leq m \end{cases} \end{split}\]

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

With the default values this represents a Kroupa (2001) IMF above $0.08M_\odot$.

Initialization

__str__(): Return str(self).

__call__(M): Calculate the value $\xi(m)$.

integrate(Mmin, Mmax)

Gives the analytical solution to the integral:

\[ \int^{M_{max}}_{M_{min}} \xi(m)dm = \xi_0 \frac{M_{max}^{\alpha_1+1} - M_{min}^{\alpha_1+1}}{\alpha + 1} + \xi_0 \xi_\textrm{continuity} \frac{M_{max}^{\alpha_2+1} - M_{min}^{\alpha_2+1}}{\alpha + 1}, \]

Note

The above equation doesn’t take into account that Mmin and Mmax could both be above/below the transition mass, but the function call does.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax)

Gives the analytical solution to the integral:

\[ \int^{M_{max}}_{M_{min}} \xi(m)dm = \xi_0 \frac{M_{max}^{\alpha_1+2} - M_{min}^{\alpha_1+2}}{\alpha + 2} + \xi_0 \xi_\textrm{continuity} \frac{M_{max}^{\alpha_2+2} - M_{min}^{\alpha_2+2}}{\alpha + 2}, \]

Note

The above equation doesn’t take into account that Mmin and Mmax could both be above/below the transition mass, but the function call does.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.L3IMF(alpha=2.3, beta=1.4, mu=0.2, normalisation='number', normalisation_value=1, Mmin=0.1, Mmax=100)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement an L3 IMF of the form

\[ \xi(m)dm = \xi_0 \left(\frac{m}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{m}{\mu}\right)^{1-\alpha}\right)^{-\beta} dm, \]

introduced in Maschberger (2013).

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

With the default parameter values of 2.3, 1.4, 0.2 this represents a best fit to Kroupa/Charier IMFs, provided in Maschberger (2013).

Initialization

__str__(): Return str(self).

_create_B_closure(a, b): Create a closure for the beta function as in Maschberger (2013), so we don’t need to carry around a, b params.

_G(M)

Auxilliary function from Maschberger (2013) used in other calculations.

Has the form

\[ G(m) = \left[1 + \left(\frac{m}{\mu}\right)^{1-\alpha} \right]^{1-\beta} \]

__call__(M): Calculate the value $\xi(m)$.

integrate(Mmin, Mmax): Gives the analytical solution to the integral: $\int^{M_{max}}{M{min}} \xi(m)dm = .

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax)

Gives the analytical solution to the integral: $\int^{M_{max}}{M{min}} m\xi(m)dm = …

From equation 25 of Maschberger (2013).

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1. Mmin : int or float, optional The minimum mass to integrate from, by default None, which means integrate from the minimum mass of normalisation.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.LognormalIMF(mc, sigma=1, normalisation='number', normalisation_value=1, Mmin=1, Mmax=500)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement a Lognormal IMF of the form

\[ \xi(m)dm = \xi_0\frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right) \]

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

Default values are not to be trusted blindly when considering Population III! The IMF is highly uncertain and you will need to make a decision.

Initialization

__str__(): Return str(self).

__call__(M): Calculate the value $\xi(m)$.

integrate(Mmin, Mmax): Gives the analytical[1] solution to the integral: $\int^{M_{max}}_{M_{min}} \xi(m)dm = FILL IN$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax): Gives the analytical[1] solution to the integral: $\int^{M_{max}}_{M_{min}} m\xi(m)dm = FILL IN$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.GeneralisedGammaIMF(alpha=-2.3, beta=-1.6, mc=10, normalisation='number', normalisation_value=1, Mmin=0.1, Mmax=100)

Bases: pimf.initialmassfunction.InitialMassFunction

Implement a Generalised Gamma Function IMF of the form $\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]$.

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

The default values of alpha, beta, mc come from Brauer+ (2025).

Initialization

static Γ_lower_incomplete(n_plus_1, u)

Lower incomplete gamma function for a given exponent.

Defined as $\Gamma(n+1, u) = \int_0^u t^n e^{-t}dt$.

__str__(): Return str(self).

__call__(M): Calculate the value $\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]$.

integrate(Mmin, Mmax)

Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} \xi(m)dm = \xi_0 \frac{M_c^{\alpha+1}}{\beta}\left[\Gamma(n+1, u_{max}) - \Gamma(n+1, u_{min})\right]$.

Where $\Gamma(n+1, u) = \int_0^u t^n e^{-t}dt$ is the lower incomplete gamma function, $n+1 = \frac{\alpha+1}{\beta}$, and $u_X = \left(\frac{M_X}{M_c}\right)^{\beta}$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

integrate_product(Mmin, Mmax)

Gives the analytical solution to the integral: $\int^{M_{max}}_{M_{min}} \xi(m)dm = \xi_0 \frac{M_c^{\alpha+2}}{\beta}\left[\Gamma(n+1, u_{max}) - \Gamma(n+1, u_{min})\right]$.

Where $\Gamma(n+1, u) = \int_0^u t^n e^{-t}dt$ is the lower incomplete gamma function, $n+1 = \frac{\alpha+2}{\beta}$, and $u_X = \left(\frac{M_X}{M_c}\right)^{\beta}$.

Warning

For performance reasons we do not check Mmin < Mmax, or are within the bounds provided for normalisation.

inverse_cdf(x)

Calculate the inverse cumulative distribution function for this IMF.

MATHS

This is intended to be used for stochastically sampling stars from the IMF using LINK

Parameters

x : int, float, or numpy array The value of the CDF at a given mass. Between 0 and 1.

Returns

type of x The mass(es) that give the x value(s) in the CDF.

class pimf.initialmassfunction.ExponentialCutoffPowerLawIMF(*args, **kwargs)

Bases: pimf.initialmassfunction.GeneralisedGammaIMF

Implement a Generalised Gamma Function IMF of the form $\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]$.

Normalisation value provides the value we are normalising to, i.e. if you want the IMF to represent 100 solar masses.

The default values of alpha, beta, mc come from Brauer+ (2025).

Initialization

pimf.initialmassfunction.popII_IMFs()

pimf.initialmassfunction.popIII_IMFs()

pimf.initialmassfunction.main()

`InitialMassFunction`
`PowerLawIMF`	Implement a power law IMF of the form \(\xi(m)dm = \xi_0 m^\alpha dm\).
`ChabrierIMF`	Implement a Chabrier IMF of the form \[\begin{split} \xi(m)dm = \xi_0 \begin{cases} \frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right), & m \leq 1M_\odot\\ \xi_\textrm{continuity} m^{\alpha}, & 1M_\odot \leq m \end{cases} \end{split}\]
`BrokenPowerLawIMF`	Implement a broken power law IMF of the form: \[\begin{split} \xi(m)dm = \xi_0\begin{cases} m^{\alpha_1}, & m \leq M_\textrm{transition} \\ \xi_\textrm{continuity}m^{\alpha_2}, & M_\textrm{transition} \leq m \end{cases} \end{split}\]
`L3IMF`	Implement an L3 IMF of the form \[ \xi(m)dm = \xi_0 \left(\frac{m}{\mu}\right)^{-\alpha}\left(1 + \left(\frac{m}{\mu}\right)^{1-\alpha}\right)^{-\beta} dm, \] introduced in Maschberger (2013).
`LognormalIMF`	Implement a Lognormal IMF of the form \[ \xi(m)dm = \xi_0\frac{1}{m}\exp\left(\frac{-\left(\log m - \log m_c\right)^2}{2\sigma^2}\right) \]
`GeneralisedGammaIMF`	Implement a Generalised Gamma Function IMF of the form \(\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]\).
`ExponentialCutoffPowerLawIMF`	Implement a Generalised Gamma Function IMF of the form \(\xi(m) = \xi_0 m^\alpha\exp\left[-\left(\frac{m}{m_c}\right)^\beta\right]\).