def propagate_astrometry(self, phi, theta, parallax, muphistar, mutheta, vrad, t0, t1):
        """
        Propagate the position of a source from the reference epoch t0 to the new epoch t1.

        Parameters
        ----------

        phi : float
            Longitude at reference epoch (radians).
        theta : float
            Latitude at reference epoch (radians).
        parallax : float
            Parallax at the reference epoch (mas).
        muphistar : float
            Proper motion in longitude (including cos(latitude) term) at reference epoch (mas/yr).
        mutheta : float
            Proper motion in latitude at reference epoch (mas/yr).
        vrad : float
            Radial velocity at reference epoch (km/s).
        t0 : float
            Reference epoch (Julian years).
        t1 : float
            New epoch (Julian years).

        Returns
        -------

        Astrometric parameters, including the "radial proper motion" (NOT the radial velocity), at the new epoch.
        phi1, theta1, parallax1, muphistar1, mutheta1, mur1 = epoch_prop_pos(..., t0, t1)
        """

        t = t1-t0
        p0, q0, r0 = normalTriad(phi, theta)

        # Convert input data to units of radians and Julian year. Use ICRS coordinate names internally to
        # avoid errors in translating the formulae to code.
        pmra0 = muphistar*self.mastorad
        pmdec0 = mutheta*self.mastorad
        plx0 = parallax*self.mastorad
        pmr0 = vrad*parallax/auKmYearPerSec*self.mastorad
        pmtot0sqr = (muphistar**2 + mutheta**2) * self.mastorad**2

        # Proper motion vector
        pmvec0 = pmra0*p0+pmdec0*q0

        f = (1 + 2*pmr0*t + (pmtot0sqr+pmr0**2)*t**2)**(-0.5)
        u = (r0*(1+pmr0*t) + pmvec0*t)*f

        _, phi1, theta1 = cartesianToSpherical(u[0], u[1], u[2])
        parallax1 = parallax*f
        pmr1 = (pmr0+(pmtot0sqr + pmr0**2)*t)*f**2
        pmvec1 = (pmvec0*(1+pmr0*t) - r0*pmr0**2*t)*f**3
        p1, q1, r1 = normalTriad(phi1, theta1)
        muphistar1 = sum(p1*pmvec1/self.mastorad, axis=0)
        mutheta1 = sum(q1*pmvec1/self.mastorad, axis =0)
        murad1 = pmr1/self.mastorad

        return phi1, theta1, parallax1, muphistar1, mutheta1, murad1
def propagate_astrometry(self, phi, theta, parallax, muphistar, mutheta, vrad, t0, t1):
        """
        Propagate the position of a source from the reference epoch t0 to the new epoch t1.

        Parameters
        ----------

        phi : float
            Longitude at reference epoch (radians).
        theta : float
            Latitude at reference epoch (radians).
        parallax : float
            Parallax at the reference epoch (mas).
        muphistar : float
            Proper motion in longitude (including cos(latitude) term) at reference epoch (mas/yr).
        mutheta : float
            Proper motion in latitude at reference epoch (mas/yr).
        vrad : float
            Radial velocity at reference epoch (km/s).
        t0 : float
            Reference epoch (Julian years).
        t1 : float
            New epoch (Julian years).

        Returns
        -------

        Astrometric parameters, including the "radial proper motion" (NOT the radial velocity), at the new epoch.
        phi1, theta1, parallax1, muphistar1, mutheta1, mur1 = epoch_prop_pos(..., t0, t1)
        """

        t = t1-t0
        p0, q0, r0 = normalTriad(phi, theta)

        # Convert input data to units of radians and Julian year. Use ICRS coordinate names internally to
        # avoid errors in translating the formulae to code.
        pmra0 = muphistar*self.mastorad
        pmdec0 = mutheta*self.mastorad
        plx0 = parallax*self.mastorad
        pmr0 = vrad*parallax/auKmYearPerSec*self.mastorad
        pmtot0sqr = (muphistar**2 + mutheta**2) * self.mastorad**2

        # Proper motion vector
        pmvec0 = pmra0*p0+pmdec0*q0

        f = (1 + 2*pmr0*t + (pmtot0sqr+pmr0**2)*t**2)**(-0.5)
        u = (r0*(1+pmr0*t) + pmvec0*t)*f

        _, phi1, theta1 = cartesianToSpherical(u[0], u[1], u[2])
        parallax1 = parallax*f
        pmr1 = (pmr0+(pmtot0sqr + pmr0**2)*t)*f**2
        pmvec1 = (pmvec0*(1+pmr0*t) - r0*pmr0**2*t)*f**3
        p1, q1, r1 = normalTriad(phi1, theta1)
        muphistar1 = sum(p1*pmvec1/self.mastorad, axis=0)
        mutheta1 = sum(q1*pmvec1/self.mastorad, axis =0)
        murad1 = pmr1/self.mastorad

        return phi1, theta1, parallax1, muphistar1, mutheta1, murad1
def propagate_astrometry ( self , phi , theta , parallax , muphistar , mutheta , vrad , t0 , t1 ) : t = t1 - t0 p0 , q0 , r0 = normalTriad ( phi , theta ) # Convert input data to units of radians and Julian year. Use ICRS coordinate names internally to # avoid errors in translating the formulae to code. pmra0 = muphistar * self . mastorad pmdec0 = mutheta * self . mastorad plx0 = parallax * self . mastorad pmr0 = vrad * parallax / auKmYearPerSec * self . mastorad pmtot0sqr = ( muphistar ** 2 + mutheta ** 2 ) * self . mastorad ** 2 # Proper motion vector pmvec0 = pmra0 * p0 + pmdec0 * q0 f = ( 1 + 2 * pmr0 * t + ( pmtot0sqr + pmr0 ** 2 ) * t ** 2 ) ** ( - 0.5 ) u = ( r0 * ( 1 + pmr0 * t ) + pmvec0 * t ) * f _ , phi1 , theta1 = cartesianToSpherical ( u [ 0 ] , u [ 1 ] , u [ 2 ] ) parallax1 = parallax * f pmr1 = ( pmr0 + ( pmtot0sqr + pmr0 ** 2 ) * t ) * f ** 2 pmvec1 = ( pmvec0 * ( 1 + pmr0 * t ) - r0 * pmr0 ** 2 * t ) * f ** 3 p1 , q1 , r1 = normalTriad ( phi1 , theta1 ) muphistar1 = sum ( p1 * pmvec1 / self . mastorad , axis = 0 ) mutheta1 = sum ( q1 * pmvec1 / self . mastorad , axis = 0 ) murad1 = pmr1 / self . mastorad return phi1 , theta1 , parallax1 , muphistar1 , mutheta1 , murad1
Propagate the position of a source from the reference epoch t0 to the new epoch t1.

        Parameters
        ----------

        phi : float
            Longitude at reference epoch (radians).
        theta : float
            Latitude at reference epoch (radians).
        parallax : float
            Parallax at the reference epoch (mas).
        muphistar : float
            Proper motion in longitude (including cos(latitude) term) at reference epoch (mas/yr).
        mutheta : float
            Proper motion in latitude at reference epoch (mas/yr).
        vrad : float
            Radial velocity at reference epoch (km/s).
        t0 : float
            Reference epoch (Julian years).
        t1 : float
            New epoch (Julian years).

        Returns
        -------

        Astrometric parameters, including the "radial proper motion" (NOT the radial velocity), at the new epoch.
        phi1, theta1, parallax1, muphistar1, mutheta1, mur1 = epoch_prop_pos(..., t0, t1)
Propagate the position of a source from the reference epoch t0 to the new epoch t1 .