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Accretion

Steady-state protostellar accretion models.

PowerLawAccrete

Tapered power-law accretion model for a protostar.

Implements the steady-state accretion law relating a protostar's instantaneous accretion rate to its current mass m and final mass mf, following the formalism of McKee & Offner (2010) and Offner & McKee (2011).

Parameters:

Name Type Description Default
j float

Power-law index of the accretion rate's dependence on m/mf.

 required
jf float

Power-law index of the accretion rate's dependence on mf.

 required
m0 float

Normalization of the accretion rate, in Msun/yr.

 required
deltan1 float

Tapering parameter that drives the accretion rate to zero as m -> mf. Default is 0 (no tapering).

 0

Attributes:

Name Type Description
j float

See Parameters.
jf float

See Parameters.
m0 float

See Parameters.
deltan1 float

See Parameters.
Source code in src/ocotillopmf/accretion.py 
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class PowerLawAccrete:
    """Tapered power-law accretion model for a protostar.

    Implements the steady-state accretion law relating a protostar's
    instantaneous accretion rate to its current mass `m` and final mass
    `mf`, following the formalism of McKee & Offner (2010) and Offner &
    McKee (2011).

    Parameters
    ----------
    j : float
        Power-law index of the accretion rate's dependence on `m/mf`.
    jf : float
        Power-law index of the accretion rate's dependence on `mf`.
    m0 : float
        Normalization of the accretion rate, in Msun/yr.
    deltan1 : float, optional
        Tapering parameter that drives the accretion rate to zero as
        `m -> mf`. Default is 0 (no tapering).

    Attributes
    ----------
    j : float
        See Parameters.
    jf : float
        See Parameters.
    m0 : float
        See Parameters.
    deltan1 : float
        See Parameters.
    """

    deltan1 = 0.0
    j = 1.0
    jf = 0.0
    m0 = 1e-5
    ml = 0.033
    mmax = 100.0

    def __init__(self, j: float, jf: float, m0: float, deltan1: float = 0) -> None:
        self.j = j
        self.jf = jf
        self.m0 = m0
        self.deltan1 = deltan1

    def acc(self, m: ArrayLike, mf: ArrayLike) -> float | np.ndarray:
        """Instantaneous mass accretion rate.

        Parameters
        ----------
        m : float or array_like
            Current protostellar mass, in Msun.
        mf : float or array_like
            Final protostellar mass, in Msun.

        Returns
        -------
        float or ndarray
            Accretion rate dm/dt, in Msun/yr.
        """
        return (
            self.m0
            * (m / mf) ** self.j
            * mf**self.jf
            * (1.0 - self.deltan1 * (m / mf) ** (1.0 - self.j)) ** (0.5)
        )

    def tm(self, mf: ArrayLike) -> float | np.ndarray:
        """Total formation timescale for a star of final mass `mf`.

        Parameters
        ----------
        mf : float or array_like
            Final protostellar mass, in Msun.

        Returns
        -------
        float or ndarray
            Formation timescale, in yr.
        """
        return (mf ** (1.0 - self.jf) / ((1.0 - self.j) * self.m0)) * (1 + self.deltan1)

    def tacc(self, m: ArrayLike, mf: ArrayLike) -> float | np.ndarray:
        """Instantaneous accretion timescale, ``m / acc(m, mf)``.

        Parameters
        ----------
        m : float or array_like
            Current protostellar mass, in Msun.
        mf : float or array_like
            Final protostellar mass, in Msun.

        Returns
        -------
        float or ndarray
            Accretion timescale, in yr.
        """
        # return (1.-self.j)*(m/mf)**(1.-self.j)*(1. - self.deltan1*(m/mf)**(1.-self.j))**(-0.5)*self.tm(mf)/(1.+self.deltan1)
        return m / self.acc(m, mf)

    def tmav(self, IMF: Callable[[float], float], ML: float, MU: float) -> float:
        """IMF-averaged formation timescale over a mass range.

        Parameters
        ----------
        IMF : callable
            Initial mass function, called as ``IMF(m)`` for a final mass
            `m`.
        ML : float
            Lower mass bound of the integral, in Msun.
        MU : float
            Upper mass bound of the integral, in Msun.

        Returns
        -------
        float
            IMF-weighted average formation timescale, in yr.
        """
        mfs = np.logspace(np.log10(ML), np.log10(MU), int(1e3))
        tmi = self.tm(mfs)
        imfi = np.array([IMF(mi) for mi in mfs])
        integrand = tmi * imfi
        return sint.trapezoid(integrand, x=np.log(mfs))

acc(m, mf)

Instantaneous mass accretion rate.

Parameters:

Name Type Description Default
m float or array_like

Current protostellar mass, in Msun.

 required
mf float or array_like

Final protostellar mass, in Msun.

 required

Returns:

Type Description
float or ndarray

Accretion rate dm/dt, in Msun/yr.
Source code in src/ocotillopmf/accretion.py 
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def acc(self, m: ArrayLike, mf: ArrayLike) -> float | np.ndarray:
    """Instantaneous mass accretion rate.

    Parameters
    ----------
    m : float or array_like
        Current protostellar mass, in Msun.
    mf : float or array_like
        Final protostellar mass, in Msun.

    Returns
    -------
    float or ndarray
        Accretion rate dm/dt, in Msun/yr.
    """
    return (
        self.m0
        * (m / mf) ** self.j
        * mf**self.jf
        * (1.0 - self.deltan1 * (m / mf) ** (1.0 - self.j)) ** (0.5)
    )

tm(mf)

Total formation timescale for a star of final mass mf.

Parameters:

Name Type Description Default
mf float or array_like

Final protostellar mass, in Msun.

 required

Returns:

Type Description
float or ndarray

Formation timescale, in yr.
Source code in src/ocotillopmf/accretion.py 
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def tm(self, mf: ArrayLike) -> float | np.ndarray:
    """Total formation timescale for a star of final mass `mf`.

    Parameters
    ----------
    mf : float or array_like
        Final protostellar mass, in Msun.

    Returns
    -------
    float or ndarray
        Formation timescale, in yr.
    """
    return (mf ** (1.0 - self.jf) / ((1.0 - self.j) * self.m0)) * (1 + self.deltan1)

tacc(m, mf)

Instantaneous accretion timescale, m / acc(m, mf).

Parameters:

Name Type Description Default
m float or array_like

Current protostellar mass, in Msun.

 required
mf float or array_like

Final protostellar mass, in Msun.

 required

Returns:

Type Description
float or ndarray

Accretion timescale, in yr.
Source code in src/ocotillopmf/accretion.py 
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def tacc(self, m: ArrayLike, mf: ArrayLike) -> float | np.ndarray:
    """Instantaneous accretion timescale, ``m / acc(m, mf)``.

    Parameters
    ----------
    m : float or array_like
        Current protostellar mass, in Msun.
    mf : float or array_like
        Final protostellar mass, in Msun.

    Returns
    -------
    float or ndarray
        Accretion timescale, in yr.
    """
    # return (1.-self.j)*(m/mf)**(1.-self.j)*(1. - self.deltan1*(m/mf)**(1.-self.j))**(-0.5)*self.tm(mf)/(1.+self.deltan1)
    return m / self.acc(m, mf)

tmav(IMF, ML, MU)

IMF-averaged formation timescale over a mass range.

Parameters:

Name Type Description Default
IMF callable

Initial mass function, called as IMF(m) for a final mass m.

 required
ML float

Lower mass bound of the integral, in Msun.

 required
MU float

Upper mass bound of the integral, in Msun.

 required

Returns:

Type Description
float

IMF-weighted average formation timescale, in yr.
Source code in src/ocotillopmf/accretion.py 
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def tmav(self, IMF: Callable[[float], float], ML: float, MU: float) -> float:
    """IMF-averaged formation timescale over a mass range.

    Parameters
    ----------
    IMF : callable
        Initial mass function, called as ``IMF(m)`` for a final mass
        `m`.
    ML : float
        Lower mass bound of the integral, in Msun.
    MU : float
        Upper mass bound of the integral, in Msun.

    Returns
    -------
    float
        IMF-weighted average formation timescale, in yr.
    """
    mfs = np.logspace(np.log10(ML), np.log10(MU), int(1e3))
    tmi = self.tm(mfs)
    imfi = np.array([IMF(mi) for mi in mfs])
    integrand = tmi * imfi
    return sint.trapezoid(integrand, x=np.log(mfs))