The star field¶

The named stars and a seeded synthetic field, by right ascension, declination and magnitude.

Requires

the world's Seed (an integer), the field's Count of synthetic stars and the Faintest magnitude it reaches.

Provides

each star's RightAscension and Declination (rad, J2000), its Magnitude (V) and its name ("" for a synthetic star), the named stars first; and each synthetic star's galactic latitude (degrees).

Law

the synthetic stars crowd the galactic plane, sin b = s u^3 (half within about 7 degrees of it), their longitude weighted towards the galactic centre, and their magnitudes follow the rising count of fainter stars, N(<m) as 10^(0.5 m); galactic to equatorial by the IAU pole. Each star is drawn from lusine.error's stream by the seed and its index, so a seed is one sky on every machine.

Source

the Yale Bright Star Catalogue, 5th ed. (the named stars); Blaauw et al., MNRAS 121 (1960) (the galactic pole); Allen's Astrophysical Quantities, 4th ed., ch. 17 (star counts).

Proof

invariant, reference, limit.

No Trait form: a tool, not a step law: a catalogue drawn once from the world's seed, the named stars and a synthetic field; it runs where a game prepares its data, never in a step, so it has no Trait form.

sky.stars(seed, count, faintest)

Uses, without proving it here: sky.horizontal, sky.sidereal.

The named stars of the Yale catalogue and a synthetic field drawn from the world's seed, crowding the galactic plane.

In [1]:

import os, sys
root = os.getcwd()
while not os.path.isfile(os.path.join(root, 'docsite.json')):
    root = os.path.dirname(root)
sys.path.insert(0, os.path.join(root, 'tools'))
import numpy as np
import matplotlib.pyplot as plt
from lusine import library
from lusine.kernels.sky._astro import deg, julian_day, to_deg
from lusine.kernels.sky.horizontal import horizontal
from lusine.kernels.sky.sidereal import sidereal
from lusine.kernels.sky.stars import CONSTELLATIONS, NAMED, stars

import os, sys root = os.getcwd() while not os.path.isfile(os.path.join(root, 'docsite.json')): root = os.path.dirname(root) sys.path.insert(0, os.path.join(root, 'tools')) import numpy as np import matplotlib.pyplot as plt from lusine import library from lusine.kernels.sky._astro import deg, julian_day, to_deg from lusine.kernels.sky.horizontal import horizontal from lusine.kernels.sky.sidereal import sidereal from lusine.kernels.sky.stars import CONSTELLATIONS, NAMED, stars

The reference¶

The kernel's reference, in numpy: the body under the declaration above, and what every section below runs. It stays as the kernel's test beside its Trait form.

In [2]:

library.reference(stars)

library.reference(stars)

Out[2]:

the reference, stars.py

def stars(seed, count=3000, faintest=6.5):
    """THE STAR FIELD: the named stars and a seeded synthetic field, by right
    ascension, declination and magnitude.

    Requires: the world's `Seed` (an integer), the field's `Count` of
    synthetic stars and the `Faintest` magnitude it reaches.
    Provides: each star's `RightAscension` and `Declination` (rad, J2000),
    its `Magnitude` (V) and its name ("" for a synthetic star), the named
    stars first; and each synthetic star's galactic latitude (degrees).
    Law: the synthetic stars crowd the galactic plane, sin b = s u^3 (half
    within about 7 degrees of it), their longitude weighted towards the
    galactic centre, and their magnitudes follow the rising count of fainter
    stars, N(<m) as 10^(0.5 m); galactic to equatorial by the IAU pole. Each
    star is drawn from lusine.error's stream by the seed and its index, so a
    seed is one sky on every machine.
    Source: the Yale Bright Star Catalogue, 5th ed. (the named stars);
    Blaauw et al., MNRAS 121 (1960) (the galactic pole); Allen's
    Astrophysical Quantities, 4th ed., ch. 17 (star counts).
    Proof: invariant, reference, limit."""
    i = np.arange(count)
    u1 = np.array([E.uniform(seed, int(k), "sky-b", 0) for k in i])
    u2 = np.array([E.uniform(seed, int(k), "sky-l", 0) for k in i])
    u3 = np.array([E.uniform(seed, int(k), "sky-m", 0) for k in i])
    sign = np.where(u1 < 0.5, -1.0, 1.0)
    sb = sign * ((np.abs(u1 - 0.5) * 2.0) ** 3)
    b = to_deg(D.dasin(sb))
    l = _wrap360(180.0 * (2.0 * u2 - 1.0) * np.abs(2.0 * u2 - 1.0))          # denser towards l = 0
    lo = 3.0
    # N(<m) as 10^(0.5 m): m = lo + 2 log10(1 + u (10^(0.5 (faintest - lo)) - 1)), log10 through dlog
    mag = lo + 2.0 * D.dlog(1.0 + u3 * (10.0 ** (0.5 * (faintest - lo)) - 1.0)) / LN10
    ra, dec = galactic_to_equatorial(l, b)
    names = list(NAMED)
    nra = deg([NAMED[n][0] for n in names])
    ndec = deg([NAMED[n][1] for n in names])
    nmag = np.array([NAMED[n][2] for n in names])
    return {"ra": np.concatenate([nra, ra]), "dec": np.concatenate([ndec, dec]), "mag": np.concatenate([nmag, mag]),
            "names": names + [""] * count, "galactic_b": np.concatenate([np.full(len(names), np.nan), b])}

Invariant: a seed is one sky¶

The same seed gives the same stars, star for star, and another seed another; the synthetic field crowds the galactic plane, most of it within 20 degrees.

In [3]:

a, b, c = stars(7, 2000), stars(7, 2000), stars(8, 2000)
assert np.array_equal(a['ra'], b['ra']) and np.array_equal(a['mag'], b['mag'])
assert not np.array_equal(a['ra'], c['ra'])
lat = a['galactic_b'][len(NAMED):]
near = float(np.mean(np.abs(lat) < 20.0))
assert near > 0.6, near
print(f'{100 * near:.0f}% of the field within 20 degrees of the galactic plane')
fig, ax = plt.subplots(figsize=(7, 3))
ax.hist(lat, bins=36, range=(-90, 90), color='k')
ax.set_xlabel('galactic latitude (degrees)')
ax.set_ylabel('stars')
plt.show()

a, b, c = stars(7, 2000), stars(7, 2000), stars(8, 2000) assert np.array_equal(a['ra'], b['ra']) and np.array_equal(a['mag'], b['mag']) assert not np.array_equal(a['ra'], c['ra']) lat = a['galactic_b'][len(NAMED):] near = float(np.mean(np.abs(lat) < 20.0)) assert near > 0.6, near print(f'{100 * near:.0f}% of the field within 20 degrees of the galactic plane') fig, ax = plt.subplots(figsize=(7, 3)) ax.hist(lat, bins=36, range=(-90, 90), color='k') ax.set_xlabel('galactic latitude (degrees)') ax.set_ylabel('stars') plt.show()

69% of the field within 20 degrees of the galactic plane

Reference: the named stars where the catalogue puts them¶

Orion's belt on the celestial equator, Polaris within a degree of the pole, Sirius the brightest star in the sky.

In [4]:

f = stars(0, 0)
names = f['names']
belt = [names.index(n) for n in CONSTELLATIONS["Orion's belt"]]
assert np.all(np.abs(to_deg(f['dec'][belt])) < 2.0)
assert 89.0 < float(to_deg(f['dec'][names.index('Polaris')])) < 90.0
assert names[int(np.argmin(f['mag']))] == 'Sirius'
print('belt declinations', np.round(to_deg(f['dec'][belt]), 2))

f = stars(0, 0) names = f['names'] belt = [names.index(n) for n in CONSTELLATIONS["Orion's belt"]] assert np.all(np.abs(to_deg(f['dec'][belt])) < 2.0) assert 89.0 < float(to_deg(f['dec'][names.index('Polaris')])) < 90.0 assert names[int(np.argmin(f['mag']))] == 'Sirius' print('belt declinations', np.round(to_deg(f['dec'][belt]), 2))

belt declinations [-1.94 -1.2  -0.3 ]

Limit: at the pole the stars neither rise nor set¶

At the north pole a star's altitude is its declination, at every hour.

In [5]:

f = stars(0, 300)
for hour in (0.0, 6.0, 13.0):
    jd = julian_day(2024, 1, 1, hour)
    alt, az = horizontal(f['ra'], f['dec'], deg(90.0), sidereal(jd, 0.0))
    assert np.allclose(alt, f['dec'], atol=1e-9)
print('fixed altitudes at the pole')
fig, ax = plt.subplots(figsize=(7, 3.5))
ax.scatter(to_deg(az), to_deg(alt), s=np.clip(6 - f['mag'], 0.3, 6), c='k')
ax.set_xlabel('azimuth (degrees)')
ax.set_ylabel('altitude (degrees)')
plt.show()

f = stars(0, 300) for hour in (0.0, 6.0, 13.0): jd = julian_day(2024, 1, 1, hour) alt, az = horizontal(f['ra'], f['dec'], deg(90.0), sidereal(jd, 0.0)) assert np.allclose(alt, f['dec'], atol=1e-9) print('fixed altitudes at the pole') fig, ax = plt.subplots(figsize=(7, 3.5)) ax.scatter(to_deg(az), to_deg(alt), s=np.clip(6 - f['mag'], 0.3, 6), c='k') ax.set_xlabel('azimuth (degrees)') ax.set_ylabel('altitude (degrees)') plt.show()

fixed altitudes at the pole