pymeeus.Sun — PyMeeus 0.5.11 documentation

# Source code for pymeeus.Sun

    # -*- coding: utf-8 -*-

    # PyMeeus: Python module implementing astronomical algorithms.
    # Copyright (C) 2018  Dagoberto Salazar
    #
    # This program is free software: you can redistribute it and/or modify
    # it under the terms of the GNU Lesser General Public License as published by
    # the Free Software Foundation, either version 3 of the License, or
    # (at your option) any later version.
    #
    # This program is distributed in the hope that it will be useful,
    # but WITHOUT ANY WARRANTY; without even the implied warranty of
    # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    # GNU Lesser General Public License for more details.
    #
    # You should have received a copy of the GNU Lesser General Public License
    # along with this program.  If not, see <https://www.gnu.org/licenses/>.

    from math import sin, cos, tan, atan, atan2, asin

    from pymeeus.Angle import Angle
    from pymeeus.Epoch import Epoch, JDE2000
    from pymeeus.Coordinates import (
        mean_obliquity,
        true_obliquity,
        nutation_longitude,
        ecliptical2equatorial,
    )
    from pymeeus.Earth import Earth

    """
    .. module:: Sun
       :synopsis: Module including functions regarding Sun position
       :license: GNU Lesser General Public License v3 (LGPLv3)

    .. moduleauthor:: Dagoberto Salazar
    """

    class Sun(object):
        """
        Class Sun handles the parameters related to the Sun.
        """

        def __init__(self):
            """Sun constructor.

            :returns: Sun object.
            :rtype: :py:class:`Sun`
            """

        @staticmethod
        def true_longitude_coarse(epoch):
            """This method provides the Sun's true longitude with a relatively low
            accuracy of about 0.01 degree.

            :param epoch: Epoch to compute the position of the Sun
            :type epoch: :py:class:`Epoch`

            :returns: A tuple containing the true (ecliptical) longitude (as an
                Angle object) and the radius vector in astronomical units.
            :rtype: tuple
            :raises: TypeError if input value is of wrong type.

            >>> epoch = Epoch(1992, 10, 13)
            >>> true_lon, r = Sun.true_longitude_coarse(epoch)
            >>> print(true_lon.dms_str(n_dec=0))
            199d 54' 36.0''
            >>> print(round(r, 5))
            0.99766
            """

            # First check that input values are of correct types
            if not (isinstance(epoch, Epoch)):
                raise TypeError("Invalid input type")
            # Compute the time in Julian centuries
            t = (epoch - JDE2000) / 36525.0
            # Compute the geometric mean longitude of the Sun
            l0 = 280.46646 + t * (36000.76983 + t * 0.0003032)
            l0 = Angle(l0)
            l0.to_positive()
            # Now, compute the mean anomaly of the Sun
            m = 357.52911 + t * (35999.05029 - t * 0.0001537)
            m = Angle(m)
            mrad = m.rad()
            # The eccentricity of the Earth's orbit
            e = 0.016708634 - t * (0.000042037 + t * 0.0000001267)
            # Equation of the center
            c = (
                (1.914602 - t * (0.004817 + t * 0.000014)) * sin(mrad)
                + (0.019993 - t * 0.000101) * sin(2.0 * mrad)
                + 0.000289 * sin(3.0 * mrad)
            )
            c = Angle(c)
            true_lon = l0 + c
            true_anom = m + c
            # Sun's radius vector
            r = (1.000001018 * (1.0 - e * e)) / (1.0 + e * cos(true_anom.rad()))
            return (true_lon, r)

        @staticmethod
        def apparent_longitude_coarse(epoch):
            """This method provides the Sun's apparent longitude with a relatively
            low accuracy of about 0.01 degree.

            :param epoch: Epoch to compute the position of the Sun
            :type epoch: :py:class:`Epoch`

            :returns: A tuple containing the sun_apparent (ecliptical) longitude
                (as an Angle object) and the radius vector in astronomical units.
            :rtype: tuple
            :raises: TypeError if input value is of wrong type.

            >>> epoch = Epoch(1992, 10, 13)
            >>> app_lon, r = Sun.apparent_longitude_coarse(epoch)
            >>> print(app_lon.dms_str(n_dec=0))
            199d 54' 32.0''
            >>> print(round(r, 5))
            0.99766
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # First find the true longitude
            sun = Sun()
            true_lon, r = sun.true_longitude_coarse(epoch)
            # Compute the time in Julian centuries
            t = (epoch - JDE2000) / 36525.0
            # Then correct for nutation and aberration
            omega = 125.04 - 1934.136 * t
            omega = Angle(omega)
            lambd = true_lon - 0.00569 - 0.00478 * sin(omega.rad())
            return (lambd, r)

        @staticmethod
        def apparent_rightascension_declination_coarse(epoch):
            """This method provides the Sun's apparent right ascension and
            declination with a relatively low accuracy of about 0.01 degree.

            :param epoch: Epoch to compute the position of the Sun
            :type epoch: :py:class:`Epoch`

            :returns: A tuple containing the right ascension and the declination
                (as Angle objects) and the radius vector in astronomical units.
            :rtype: tuple
            :raises: TypeError if input value is of wrong type.

            >>> epo = Epoch(1992, 10, 13)
            >>> ra, delta, r = Sun.apparent_rightascension_declination_coarse(epo)
            >>> print(ra.ra_str(n_dec=1))
            13h 13' 31.4''
            >>> print(delta.dms_str(n_dec=0))
            -7d 47' 6.0''
            >>> print(round(r, 5))
            0.99766
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Second, find the apparent longitude
            sun = Sun()
            app_lon, r = sun.apparent_longitude_coarse(epoch)
            # Compute the obliquity of the ecliptic
            e0 = mean_obliquity(epoch)
            # Compute the time in Julian centuries
            t = (epoch - JDE2000) / 36525.0
            # Then correct for nutation and aberration
            omega = 125.04 - 1934.136 * t
            omega = Angle(omega)
            # Correct the obliquity
            e = e0 + 0.00256 * cos(omega.rad())
            alpha = atan2(cos(e.rad()) * sin(app_lon.rad()), cos(app_lon.rad()))
            alpha = Angle(alpha, radians=True)
            alpha.to_positive()
            delta = asin(sin(e.rad()) * sin(app_lon.rad()))
            delta = Angle(delta, radians=True)
            return (alpha, delta, r)

        @staticmethod
        def geometric_geocentric_position(epoch, tofk5=True):
            """This method computes the geometric geocentric position of the Sun
            for a given epoch, using the VSOP87 theory.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`
            :param tofk5: Whether or not the small correction to convert to the FK5
                system will be applied or not
            :type tofk5: bool

            :returns: A tuple with the geocentric longitude and latitude (as
                :py:class:`Angle` objects), and the radius vector (as a float,
                in astronomical units), in that order
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> l, b, r = Sun.geometric_geocentric_position(epoch, tofk5=False)
            >>> print(round(l.to_positive(), 6))
            199.907297
            >>> print(b.dms_str(n_dec=3))
            0.744''
            >>> print(round(r, 8))
            0.99760852
            """

            # NOTE: In page 169, Meeus gives a different value for the LONGITUDE
            # (199.907372 degrees) as the one presented above (199.907297 degrees).
            # After many checks and tests, I came to the conclusion that the result
            # above is the right one, and Meeus' result is wrong.
            # On the other hand, the difference in LATITUDE may be due to the fact
            # that this software uses the complete set of VSOP87C terms, instead of
            # the abridged version in Meeus' book.

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch) and not isinstance(tofk5, bool):
                raise TypeError("Invalid input types")
            # Use Earth heliocentric position to compute Sun's geocentric position
            lon, lat, r = Earth.geometric_heliocentric_position(epoch, tofk5)
            lon = lon.to_positive() + 180.0
            lat = -lat
            return lon, lat, r

        @staticmethod
        def apparent_geocentric_position(epoch, nutation=True):
            """This method computes the apparent geocentric position of the Sun
            for a given epoch, using the VSOP87 theory.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`
            :param nutation: Whether the nutation correction will be applied
            :type epoch: bool

            :returns: A tuple with the heliocentric longitude and latitude (as
                :py:class:`Angle` objects), and the radius vector (as a float,
                in astronomical units), in that order
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> lon, lat, r = Sun.apparent_geocentric_position(epoch)
            >>> print(lon.to_positive().dms_str(n_dec=3))
            199d 54' 21.548''
            >>> print(lat.dms_str(n_dec=3))
            0.721''
            >>> print(round(r, 8))
            0.99760852
            """

            # NOTE: In page 169, Meeus gives a different value for the LONGITUDE
            # (199d 54' 21.818'') as the one presented above (199d 54' 21.548'').
            # After many checks and tests, I came to the conclusion that the result
            # above is the right one, and Meeus' result is wrong.
            # On the other hand, the difference in LATITUDE may be due to the fact
            # that this software uses the complete set of VSOP87C terms, instead of
            # the abridged version in Meeus' book.

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Use Earth heliocentric position to compute Sun's geocentric position
            lon, lat, r = Earth.apparent_heliocentric_position(epoch, nutation)
            lon = lon.to_positive() + 180.0
            lat = -lat
            return lon, lat, r

        @staticmethod
        def rectangular_coordinates_mean_equinox(epoch):
            """This method computes the rectangular geocentric equatorial
            coordinates (X, Y, Z) of the Sun, referred to the mean equinox of the
            date. The X axis is directed towards the vernal equinox (longitude 0),
            the Y axis lies in the plane of the equator and is directed towards
            longitude 90, and the Z axis is directed towards the north celestial
            pole.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`

            :returns: A tuple with the X, Y, Z values in astronomical units
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> x, y, z = Sun.rectangular_coordinates_mean_equinox(epoch)
            >>> print(round(x, 7))
            -0.9379963
            >>> print(round(y, 6))
            -0.311654
            >>> print(round(z, 7))
            -0.1351207
            """

            # NOTE: In page 172, Meeus gives slightly different values for x, y, z
            # as the ones internally computed in the example above. After many
            # checks and tests, I came to the conclusion that the results above are
            # the right ones, and Meeus' results are wrong.

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Get Sun's geocentric position with reduction to FK5 system
            lon, lat, r = Sun.geometric_geocentric_position(epoch)
            # Get the mean obliquity
            epsilon0 = mean_obliquity(epoch)
            # Change to radians
            ll = lon.rad()
            b = lat.rad()
            e = epsilon0.rad()
            # Compute the results
            x = r * cos(ll)
            y = r * (sin(ll) * cos(e) - sin(b) * sin(e))
            z = r * (sin(ll) * sin(e) + sin(b) * cos(e))
            return x, y, z

        @staticmethod
        def rectangular_coordinates_j2000(epoch):
            """This method computes the rectangular geocentric equatorial
            coordinates (X, Y, Z) of the Sun, referred to the standard equinox of
            J2000.0. The X axis is directed towards the vernal equinox (longitude
            0), the Y axis lies in the plane of the equator and is directed towards
            longitude 90, and the Z axis is directed towards the north celestial
            pole.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`

            :returns: A tuple with the X, Y, Z values in astronomical units
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> x, y, z = Sun.rectangular_coordinates_j2000(epoch)
            >>> print(round(x, 8))
            -0.93740485
            >>> print(round(y, 8))
            -0.3131474
            >>> print(round(z, 8))
            -0.13577045
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Second, compute Earth heliocentric position referred to J2000.0
            lon, lat, r = Earth.geometric_heliocentric_position_j2000(epoch)
            # Third, convert from Earth's heliocentric to Sun's geocentric
            lon = lon.to_positive() + 180.0
            lat = -lat
            x = r * cos(lat.rad()) * cos(lon.rad())
            y = r * cos(lat.rad()) * sin(lon.rad())
            z = r * sin(lat.rad())
            x0 = x + 0.00000044036 * y - 0.000000190919 * z
            y0 = -0.000000479966 * x + 0.917482137087 * y - 0.397776982902 * z
            z0 = 0.397776982902 * y + 0.917482137087 * z
            return x0, y0, z0

        @staticmethod
        def rectangular_coordinates_b1950(epoch):
            """This method computes the rectangular geocentric equatorial
            coordinates (X, Y, Z) of the Sun, referred to the mean equinox of
            B1950.0. The X axis is directed towards the vernal equinox (longitude
            0), the Y axis lies in the plane of the equator and is directed towards
            longitude 90, and the Z axis is directed towards the north celestial
            pole.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`

            :returns: A tuple with the X, Y, Z values in astronomical units
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> x, y, z = Sun.rectangular_coordinates_b1950(epoch)
            >>> print(round(x, 8))
            -0.94149557
            >>> print(round(y, 8))
            -0.30259922
            >>> print(round(z, 8))
            -0.11578695
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Second, compute Earth heliocentric position referred to J2000.0
            lon, lat, r = Earth.geometric_heliocentric_position_j2000(epoch)
            # Third, convert from Earth's heliocentric to Sun's geocentric
            lon = lon.to_positive() + 180.0
            lat = -lat
            x = r * cos(lat.rad()) * cos(lon.rad())
            y = r * cos(lat.rad()) * sin(lon.rad())
            z = r * sin(lat.rad())
            x = 0.999925702634 * x + 0.012189716217 * y + 0.000011134016 * z
            y = -0.011179418036 * x + 0.917413998946 * y - 0.397777041885 * z
            z = -0.004859003787 * x + 0.397747363646 * y + 0.917482111428 * z
            return x, y, z

        @staticmethod
        def rectangular_coordinates_equinox(epoch, equinox_epoch):
            """This method computes the rectangular geocentric equatorial
            coordinates (X, Y, Z) of the Sun, referred to an arbitrary mean
            equinox. The X axis is directed towards the vernal equinox (longitude
            0), the Y axis lies in the plane of the equator and is directed towards
            longitude 90, and the Z axis is directed towards the north celestial
            pole.

            :param epoch: Epoch to compute Sun position, as an Epoch object
            :type epoch: :py:class:`Epoch`
            :param equinox_epoch: Epoch corresponding to the mean equinox
            :type equinox_epoch: :py:class:`Epoch`

            :returns: A tuple with the X, Y, Z values in astronomical units
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> e_equinox = Epoch(2467616.0)
            >>> x, y, z = Sun.rectangular_coordinates_equinox(epoch, e_equinox)
            >>> print(round(x, 8))
            -0.93368986
            >>> print(round(y, 8))
            -0.32235085
            >>> print(round(z, 8))
            -0.13977098
            """

            # First check that input values are of correct types
            if (not isinstance(epoch, Epoch)
                    and not isinstance(equinox_epoch, Epoch)):
                raise TypeError("Invalid input types")
            # Second, compute Sun's rectangular coordinates w.r.t. J2000.0
            x0, y0, z0 = Sun.rectangular_coordinates_j2000(epoch)
            # Third, computed auxiliary angles
            t = (equinox_epoch - JDE2000) / 36525.0
            tt = (epoch - equinox_epoch) / 36525.0
            # Compute the conversion parameters
            zeta = t * (
                (2306.2181 + tt * (1.39656 - 0.000139 * tt))
                + t * ((0.30188 - 0.000344 * tt) + 0.017998 * t)
            )
            z = t * (
                (2306.2181 + tt * (1.39656 - 0.000139 * tt))
                + t * ((1.09468 + 0.000066 * tt) + 0.018203 * t)
            )
            theta = t * (
                2004.3109
                + tt * (-0.85330 - 0.000217 * tt)
                + t * (-(0.42665 + 0.000217 * tt) - 0.041833 * t)
            )
            # Redefine the former values as Angles, and compute them in radians
            zeta = Angle(0, 0, zeta)
            zetar = zeta.rad()
            z = Angle(0, 0, z)
            zr = z.rad()
            theta = Angle(0, 0, theta)
            thetar = theta.rad()
            xx = cos(zetar) * cos(zr) * cos(thetar) - sin(zetar) * sin(zr)
            xy = sin(zetar) * cos(zr) + cos(zetar) * sin(zr) * cos(thetar)
            xz = cos(zetar) * sin(thetar)
            yx = -cos(zetar) * sin(zr) - sin(zetar) * cos(zr) * cos(thetar)
            yy = cos(zetar) * cos(zr) - sin(zetar) * sin(zr) * cos(thetar)
            yz = -sin(zetar) * sin(thetar)
            zx = -cos(zr) * sin(thetar)
            zy = -sin(zr) * sin(thetar)
            zz = cos(thetar)
            xp = xx * x0 + yx * y0 + zx * z0
            yp = xy * x0 + yy * y0 + zy * z0
            zp = xz * x0 + yz * y0 + zz * z0
            return xp, yp, zp

        @staticmethod
        def get_equinox_solstice(year, target="spring"):
            """This method computes the times of the equinoxes or the solstices.

            :param year: Year we want to compute the equinox or solstice for
            :type year: int
            :param target: Corresponding equinox or solstice. It can be "spring",
                "summer", "autumn", "winter"
            :type target: str

            :returns: The instant of time when the equinox or solstice happens
            :rtype: :py:class:`Epoch`
            :raises: TypeError if input values are of wrong type.
            :raises: ValueError if 'target' value is invalid.

            >>> epoch = Sun.get_equinox_solstice(1962, target="summer")
            >>> y, m, d, h, mi, s = epoch.get_full_date()
            >>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
            1962/6/21 21:24:42.0
            """

            # NOTE: The results from the previous example are computed using the
            # complete VSOP87 theory. The results provided by Meeus in exercises
            # 27.a and 27.b are computed with lower accuracy

            # First check that input values are of correct types
            if not (isinstance(year, int) and isinstance(target, str)):
                raise TypeError("Invalid input types")
            # Second, check that the target is correct
            if (
                (target != "spring")
                and (target != "summer")
                and (target != "autumn")
                and (target != "winter")
            ):
                raise ValueError("'target' value is invalid")
            # Now we can start computing an approximate value (Tables 27.A, 27.B)
            if (year >= -1000) and (year < 1000):
                y = year / 1000.0
                if target == "spring":
                    jde0 = 1721139.29189 + y * (
                        365242.1374 + y * (0.06134 + y * (0.00111 - y * 0.00071))
                    )
                elif target == "summer":
                    jde0 = 1721233.25401 + y * (
                        365241.72562 + y * (-0.05323 + y * (0.00907 + y * 0.00025))
                    )
                elif target == "autumn":
                    jde0 = (1721325.70455
                            + y * (365242.49558
                                   + y * (-0.11677 + y * (-0.00297
                                                          + y * 0.00074))))
                elif target == "winter":
                    jde0 = (1721414.39987
                            + y * (365242.88257 + y * (-0.00769
                                                       + y * (-0.00933
                                                              - y * 0.00006))))
            elif (year >= 1000) and (year <= 3000):
                y = (year - 2000.0) / 1000.0
                if target == "spring":
                    jde0 = 2451623.80984 + y * (
                        365242.37404 + y * (0.05169 + y * (-0.00411 - y * 0.00057))
                    )
                elif target == "summer":
                    jde0 = 2451716.56767 + y * (
                        365241.62603 + y * (0.00325 + y * (0.00888 - y * 0.0003))
                    )
                elif target == "autumn":
                    jde0 = 2451810.21715 + y * (
                        365242.01767 + y * (-0.11575 + y * (0.00337 + y * 0.00078))
                    )
                elif target == "winter":
                    jde0 = (2451900.05952
                            + y * (365242.74049
                                   + y * (-0.06223 + y * (-0.00823
                                                          + y * 0.00032))))
            else:
                raise ValueError("'year' value out of range")
            k = ["spring", "summer", "autumn", "winter"].index(target)
            epoch = Epoch(jde0)
            corr = 1.0
            while abs(corr) > 0.0000025:
                lon, lat, r = Sun.apparent_geocentric_position(epoch)
                arg = k * 90.0 - lon.to_positive()
                arg = Angle(arg)
                corr = 58.0 * sin(arg.rad())
                epoch += corr
            epoch -= corr
            return epoch

        @staticmethod
        def equation_of_time(epoch):
            """This method computes the equation of time for a given epoch,
            understood as the difference between apparent and mean time, or the
            difference between the hour angle of the true Sun and the mean Sun.

            :param epoch: Epoch to compute the equation of time, as an Epoch object
            :type epoch: :py:class:`Epoch`

            :returns: Difference between apparent and mean time, as a tuple, in
                minutes (int) and seconds (float) of time
            :rtype: tuple
            :raises: TypeError if input values are of wrong type.

            >>> epoch = Epoch(1992, 10, 13.0)
            >>> m, s = Sun.equation_of_time(epoch)
            >>> print(m)
            13
            >>> print(round(s, 1))
            42.6
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Compute time in Julian millenia from J2000.0
            t = (epoch - JDE2000) / 365250
            l0 = (280.4664567
                  + t * (360007.6982779
                         + t * (0.03032028
                                + t * (1.0 / 49931.0
                                       + t * (-1.0 / 15300.0
                                              - t * 1.0 / 2000000.0)))))
            l0 = Angle(l0)
            l0 = l0.to_positive()
            # Compute the apparent position of the Sun
            lon, lat, r = Sun.apparent_geocentric_position(epoch)
            # Now, get the true obliquity
            epsilon = true_obliquity(epoch)
            # Transform from eclliptical to equatorial coordinates
            alpha, dec = ecliptical2equatorial(lon, lat, epsilon)
            alpha = alpha.to_positive()
            # Now we need the nutation in longitude
            deltapsi = nutation_longitude(epoch)
            e = l0() - 0.0057183 - alpha + deltapsi * cos(epsilon.rad())
            e *= 4.0
            # Extract seconds
            s = (abs(e()) % 1) * 60.0
            m = int(e())
            return m, s

        @staticmethod
        def ephemeris_physical_observations(epoch):
            """This method uses Carrington's formulas to compute the following
            quantities:

            - P  : position angle of the northern extremity of the axis of rotation
            - B0 : heliographic latitude of the center of the solar disk
            - L0 : heliographic longitude of the center of the solar disk

            :param epoch: Epoch to compute the parameters
            :type epoch: :py:class:`Epoch`

            :returns: Parameters P, B0 and L0, in a tuple
            :rtype: tuple
            :raises: TypeError if input value is of wrong type.

            >>> epoch = Epoch(1992, 10, 13)
            >>> p, b0, l0 = Sun.ephemeris_physical_observations(epoch)
            >>> print(round(p, 2))
            26.27
            >>> print(round(b0, 2))
            5.99
            >>> print(round(l0, 2))
            238.63
            """

            # First check that input values are of correct types
            if not isinstance(epoch, Epoch):
                raise TypeError("Invalid input type")
            # Compute the auxiliary parameters
            epoch += 0.00068
            theta = (epoch() - 2398220.0) * 360.0 / 25.38
            theta = Angle(theta)
            i = Angle(7.25)
            k = 73.6667 + 1.3958333 * (epoch() - 2396758.0) / 36525.0
            k = Angle(k)
            lon, lat, r = Sun.apparent_geocentric_position(epoch, nutation=False)
            eps = true_obliquity(epoch)
            dpsi = nutation_longitude(epoch)
            lonp = lon + dpsi
            x = atan(-cos(lonp.rad()) * tan(eps.rad()))
            x = Angle(x, radians=True)
            delta = lon - k
            y = atan(-cos(delta.rad()) * tan(i.rad()))
            y = Angle(y, radians=True)
            p = x + y
            b0 = asin(sin(delta.rad()) * sin(i.rad()))
            b0 = Angle(b0, radians=True)
            eta = atan(tan(delta.rad()) * cos(i.rad()))
            eta = Angle(eta, radians=True)
            l0 = eta - theta
            return p, b0, l0.to_positive()

        @staticmethod
        def beginning_synodic_rotation(number):
            """This method calculates the epoch when the Carrington's synodic
            rotation No. 'number' starts.

            :param number: Number of Carrington's synodic rotation
            :type number: int

            :returns: Epoch when the provided rotation starts
            :rtype: :py:class:`Epoch`
            :raises: TypeError if input value is of wrong type.

            >>> epoch = Sun.beginning_synodic_rotation(1699)
            >>> print(round(epoch(), 3))
            2444480.723
            """

            # First check that input values are of correct types
            if not isinstance(number, int):
                raise TypeError("Invalid input type")
            # Apply formula (29.1)
            jde = 2398140.227 + 27.2752316*number
            # Now, find the correction using formula (29.2)
            m = 281.96 + 26.882476*number
            m = Angle(m)
            m = m.rad()
            delta = 0.1454 * sin(m) - 0.0085 * sin(2.0 * m) - 0.0141 * cos(2.0 * m)
            # Apply the correction
            jde += delta
            return Epoch(jde)

    def main():

        # Let's define a small helper function
        def print_me(msg, val):
            print("{}: {}".format(msg, val))

        # Let's show some uses of Sun functions
        print("\n" + 35 * "*")
        print("*** Use of Sun class")
        print(35 * "*" + "\n")

        # Compute an approximation of the Sun's true longitude
        epoch = Epoch(1992, 10, 13)
        true_lon, r = Sun.true_longitude_coarse(epoch)
        print_me("Sun's approximate true longitude", true_lon.dms_str(n_dec=0))
        # 199d 54' 36.0''
        print_me("Sun's radius vector", round(r, 5))  # 0.99766

        print("")

        # Now let's compute the Sun's approximate apparent longitude
        app_lon, r = Sun.apparent_longitude_coarse(epoch)
        print_me("Sun's approximate apparent longitude", app_lon.dms_str(n_dec=0))
        # 199d 54' 32.0''

        print("")

        # And now is the turn for the apparent right ascension and declination
        ra, delta, r = Sun.apparent_rightascension_declination_coarse(epoch)
        print_me("Sun's apparent right ascension", ra.ra_str(n_dec=1))
        # 13h 13' 31.4''
        print_me("Sun's apparent declination", delta.dms_str(n_dec=0))
        # -7d 47' 6.0''

        print("")

        # Let's compute Sun's postion, but more accurately
        epoch = Epoch(1992, 10, 13.0)
        l, b, r = Sun.geometric_geocentric_position(epoch, tofk5=False)
        print_me("Geometric Geocentric Longitude", round(l.to_positive(), 6))
        # 199.906016
        print_me("Geometric Geocentric Latitude", b.dms_str(n_dec=3))
        # 0.644''
        print_me("Radius vector", round(r, 8))
        # 0.99760775

        print("")

        # Compute Sun's apparent postion
        l, b, r = Sun.apparent_geocentric_position(epoch)
        print_me("Apparent Geocentric Longitude", l.to_positive().dms_str(n_dec=3))
        # 199d 54' 16.937''
        print_me("Apparent Geocentric Latitude", b.dms_str(n_dec=3))
        # 0.621''
        print_me("Radius vector", round(r, 8))
        # 0.99760775

        print("")

        # We can compute rectangular coordinates referred to mean equinox of date
        x, y, z = Sun.rectangular_coordinates_mean_equinox(epoch)
        print("Rectangular coordinates referred to mean equinox of date:")
        print_me("X", round(x, 7))  # -0.9379963
        print_me("Y", round(y, 6))  # -0.311654
        print_me("Z", round(z, 7))  # -0.1351207

        print("")

        # Now, compute rectangular coordinates w.r.t. standard equinox J2000.0
        x, y, z = Sun.rectangular_coordinates_j2000(epoch)
        print("Rectangular coordinates w.r.t. standard equinox J2000.0:")
        print_me("X", round(x, 8))  # -0.93740485
        print_me("Y", round(y, 8))  # -0.3131474
        print_me("Z", round(z, 8))  # -0.12456646

        print("")

        # Compute rectangular coordinates w.r.t. mean equinox of B1950.0
        x, y, z = Sun.rectangular_coordinates_b1950(epoch)
        print("Rectangular coordinates w.r.t. mean equinox of B1950.0:")
        print_me("X", round(x, 8))  # -0.94149557
        print_me("Y", round(y, 8))  # -0.30259922
        print_me("Z", round(z, 8))  # -0.11578695

        print("")

        # Compute rectangular coordinates w.r.t. an arbitrary mean equinox
        e_equinox = Epoch(2467616.0)
        x, y, z = Sun.rectangular_coordinates_equinox(epoch, e_equinox)
        print("Rectangular coordinates w.r.t. an arbitrary mean equinox:")
        print_me("X", round(x, 8))  # -0.93373777
        print_me("Y", round(y, 8))  # -0.32235109
        print_me("Z", round(z, 8))  # -0.12856709

        print("")

        # We can compute the date of equinoxes and solstices
        epoch = Sun.get_equinox_solstice(1962, target="summer")
        y, m, d, h, mi, s = epoch.get_full_date()
        print("The summer solstice of 1962:")
        print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
        # 1962/6/21 21:24:42.0

        epoch = Sun.get_equinox_solstice(2018, target="autumn")
        y, m, d, h, mi, s = epoch.get_full_date()
        print("The autumn equinox of 2018:")
        print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
        # 2018/9/23 1:55:14.0

        print("")

        # The equation of time, i.e., the difference between apparent and mean
        # time, can be easily computed
        epoch = Epoch(1992, 10, 13.0)
        m, s = Sun.equation_of_time(epoch)
        print("Equation of time difference: {} min {} secs".format(m, round(s, 1)))
        # 13m 42.6s

        print("")

        # Compute the ephemeris of physical observations of the Sun using
        # Carrington's formulas
        epoch = Epoch(1992, 10, 13)
        p, b0, l0 = Sun.ephemeris_physical_observations(epoch)
        print("Ephemeris of physical observations of the Sun:")
        print_me("P ", round(p, 2))     # 26.27
        print_me("B0", round(b0, 2))    # 5.99
        print_me("L0", round(l0, 2))    # 238.63

        print("")

        # Get the epoch when the Carrington's synodic rotation No. 'number' starts
        epoch = Sun.beginning_synodic_rotation(1699)
        print_me("Epoch for Carrington's synodic rotation No. 1699",
                 round(epoch(), 3))     # 2444480.723

    if __name__ == "__main__":

        main()