Coordinates and time

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Coordinates and time Sections 18 23
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18. Sidereal time A time-keeping system based on
the diurnal motion of the stars, rather than the
Sun. Local sidereal time (LST) R.A. of stars
crossing observers meridian (H 0) at any
instant hour angle of ?, the First Point of
Aries. 24 sidereal hours time interval between
two successive meridian passages of a given
star, or of First Point of Aries time for Earth
to rotate through 360?.


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Diagram showing the concept of sidereal time,
being the right ascension of stars now crossing
the observers meridian, or the hour angle of
the First Point of Aries
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Note that in 24 mean solar hours Earth rotates
through nearly 361?, because direction to Sun
relative to stars changes 1?/day. In 1 year
there are 365¼ (mean solar) days but 366¼ Earth
rotations. 1 mean solar day 24 mean solar
hours 24h 04m sidereal hours. 24h
(sidereal time) 23h 56m solar time (stars
take less time than Sun between successive
transits)
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Sidereal and solar days
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Ratio
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Relation between LST, R.A. and hour angle
LST R.A. H As one
looks E from meridian, R.A. increases at rate of
1 h every 15? on equator, H decreases at same
rate (for observer at fixed location on
Earth). On meridian H 0 and LST R.A. LST
increases by 4 m every 1? in longitude that one
travels to E.
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LST is the measure of which stars are crossing
observers meridian at any instant.
Observer at LST A 0h B 6h
C 12 D 18h
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Local sidereal time, LST
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The concepts of local sidereal time and Greenwich
sidereal time. LST HA ? for a local observer
GST HA ? for an observer at Greenwich.
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19. Ecliptic coordinates Ecliptic longitude
? Ecliptic latitude ? 0? ? ? ? 360? measured
eastwards around ecliptic 90º ? ? ? 90?
angular distance from ecliptic (? 0?) towards
N or S ecliptic poles. The zero point of
ecliptic longitude is the First Point of Aries
(?) for which (?, ?) (0?, 0?).
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K is the north ecliptic pole. P is the
north celestial pole.
Ecliptic coordinates (?,ß) for an object at point
X
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20. Galactic coordinates Galactic longitude
l Galactic latitude b 0? ? l ? 360? measured
eastwards around galactic equator. 90? ? b ?
90? angular distance from galactic equator
towards N or S galactic poles. Galactic equator
based on HI distribution in disk of our Milky
Way the Galaxy. Inclined at 62? 08? (epoch
2000) to the celestial equator.
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The galactic equator, used to define the galactic
coordinate system, is in turn defined by the
distribution of HI in the Milky Way as observed
by radio telescopes working at the 21-cm
wavelength
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l ? 0?, b ? 0? is the galactic centre
direction. N galactic pole is b ? 90? NGP is at
(?, ?)2000 ? (12h 51m, 27? 08?) Galactic centre
is at
Galactic rotation direction is (l, b) ? (90?, 0?)
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Galactic coordinates L is the galactic centre, G
is the north galactic pole, G' is the south
galactic pole.
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21. Spherical geometry Spherical triangle This
is a figure formed on the surface of a sphere by
three intersecting arcs of great circles.
ABC is a spherical triangle (formed by 3 great
circle arcs).
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Angles of spherical triangle (A, B, C) measured
in degrees and equal to angle between the
tangents to the great circles that intersect at
each vertex of the triangle. Sides of the
spherical triangle (a, b, c) measured in
degrees and equal to the angle between the radii
from spheres centre to two points of
intersection of great circles.
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The construction of spherical triangles from
great circles on the surface of a sphere
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22. Spherical trigonometry
Cosine formula cos a ? cos b cos c sin b sin
c cos A Similarly cos b ? cos a cos c
sin a sin c cos B cos c ? cos a cos b sin
a sin b cos C
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Note for small a, b, c cos a (1 ? a2/2)
etc. provided a is measured in radians instead
of degrees.
The spherical ? then approximates the plane
?. Exercise Prove that a2 ? b2 c2 ?2bc cos A
for a spherical ? with small sides.
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Sine formula
Note for small a, b, c
(sine rule for plane ?s).
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• 23. Applications of spherical trigonometry
• some practical examples
• Angle between two stars of given (?, ?)

(?1, ?1) and (?2, ?2) are coordinates ?? ?2 ?
?1 in degrees (1h ? 15?) cos ?12 ? cos (90º ? ?1)
cos (90º ? ?2) sin (90º ? ?1)
sin (90º ? ?2) cos (??) ? sin ?1 sin ?2 cos
?1 cos ?2 cos (??)
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b) Relationship between (?, ?) and (a, A)
(H, ?) (hour angle, dec) equatorial
coordinates or (?, ?) ? (R.A., dec) (a, A)
alt, az
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Consider spherical ? PZ? cos (90? ? ?) ? cos
(90? ? ?) cos z sin
(90? ? ?) sin z cos (360? ? A) ? sin ? ?
sin ? sin a cos ? cos a cos A (1)
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Also cos z ? cos (90? ? ?) cos (90? ? ?)
sin (90? ?? ?) sin (90? ? ?) cos
H sin a ? sin ? sin ? cos ? cos ?
cos H (2)
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And

(3)
Any two of these 3 equations allow one to find
(a, A) from (H, ?) or (H, ?) from (a, A). If
right ascension is needed then use ? L.S.T. ? H.
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Thus if (H, ?) are known obtain sin a from (2),
hence a then get sin A from (3), hence A. If
(a, A) are known obtain sin ? from (1), hence
? obtain sin H from (3), hence H.
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c) Azimuth of setting/rising object of
declination ?
Azimuth, A, of a rising or setting object (z
90º).
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cos (90? ? ?) ? cos (90? ? ?) cos 90?
sin (90? ? ?) sin 90? cos (360? ?
A) ? sin ? ? cos ? cos A ? cos
A ? sin d / cos ?
(or cos A ? ? sin d / cos ? in S hemisphere)
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e.g. azimuth limits of setting Sun in
Christchurch ? ? 43? 31? cos ? ? 0.725 ? ?
? 23? 27? (mid summer) 23? 27? (mid
winter) a) mid-summer cos A ? 0.549
A ?
56.7? sunrise
or 303.3? sunset b) mid-winter
cos A ? ?0.549
A ? 123.3? sunrise
or 236.7? sunset
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In general for Sun on horizon in Christchurch
A ? 90? ? 33.3? at sunrise ? 270? ? 33.3?
at sunset. where the variation is the overall
seasonal range.
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d) Time of sunrise and sunset

• cos 90? ? cos (90? ? ?) cos (90? ? ?)
• sin (90? ? ?) sin (90? ? ?) cos H
• ? 0 ? sin ? sin ? cos ? cos ? cos H
• cos H ? ? tan ? tan ?
• (or tan ? tan ? in S hemisphere)
• This gives H, the hour angle of rising/setting
  Sun,
• which is approximately the time interval between
  noon
• and sunrise/set.

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Example length of daylight hours in
Christchurch ? ? 43? 31? tan ? ?
0.950 mid-summer cos H ? ?0.412 H ? ?114.3? ?
?7.62 h length of day ? 2H ? 15.24 h ? 15 h 15
m mid-winter cos H ? 0.412 H ? ?65.68? ?
?4.38 h length of day ? 2H ? 8.76 h ? 8 h 45 m
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e) Equinoctial corollaries We have (from 23c)
cos A ? sin ?/cos ? At equinox
Sun is on equator, ? ? 0? ? sin ? ? 0 ? cos A ? 0
or A ? 90?, 270? ? At equinoxes, Sun rises due
E, sets due W (irrespective of observers
latitude).
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Also (from 23d) cos H ? ?tan ? tan ? At
equinox, tan ? ? 0 ? cos H ? 0 or H
? 90?, 270? ? 6 h, 18 h (? ? 6 h)
(as 1 h ? 15?) Length of day at equinox ? 2H
? 12 h
? length of night (equinox ? equal day and
night). Once again, this result is independent
of ?, the observers latitude.
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End of sections 18 - 23