Transcript Local sidereal time

ASTR211: COORDINATES AND TIME
Coordinates and time
Prof. John Hearnshaw
Sections 18 – 23
ASTR211: COORDINATES AND TIME
18. Sidereal time
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.
Prof. John Hearnshaw
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
observer’s meridian (H = 0) at any instant
= hour angle of , the First Point of Aries.
Diagram showing the concept of sidereal time, being the
right ascension of stars now crossing the observer’s
meridian, or the hour angle of the First Point of Aries
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
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 (sidereal time)
= 23h 56m solar time
(stars take less time than Sun between successive
transits)
Prof. John Hearnshaw
= 24h 04m sidereal hours.
Sidereal and solar days
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
Ratio
Prof. John Hearnshaw
solar day
366.25

 1.00274
sidereal day 365.25
ASTR211: COORDINATES AND TIME
Relation between LST, R.A. and hour angle
LST = R.A. + H
On meridian H = 0 and LST = R.A.
LST increases by 4 m every 1 in longitude that
one travels to E.
Prof. John Hearnshaw
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).
ASTR211: COORDINATES AND TIME
LST is the measure of which stars are crossing
observer’s meridian at any instant.
RA=6h
B
Observer at
A
B
C
D
C
LST
0h
6h
12
18h
Earth
NP

A
RA=0h
D
RA=18h
Prof. John Hearnshaw
RA=12h
Local sidereal time, LST
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
The concepts of local sidereal time and Greenwich
sidereal time. LST = HA  for a local observer;
GST = HA  for an observer at Greenwich.
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
19. Ecliptic coordinates
The zero point of ecliptic longitude is the
First Point of Aries () – for which (, ) = (0, 0).
Prof. John Hearnshaw
Ecliptic longitude 
Ecliptic latitude 
0    360 measured eastwards around ecliptic
–90º    + 90 angular distance from ecliptic ( = 0)
towards N or S ecliptic poles.
ASTR211: COORDINATES AND TIME
K is the north
ecliptic pole.
Ecliptic coordinates (λ,β) for an object at point X
Prof. John Hearnshaw
P is the north
celestial pole.
ASTR211: COORDINATES AND TIME
20. Galactic coordinates
Galactic longitude l
Galactic latitude b
Galactic equator based on HI distribution in disk of
our Milky Way – the Galaxy.
Inclined at 62 08 (epoch 2000) to the celestial equator.
Prof. John Hearnshaw
0  l  360 measured eastwards around galactic equator.
–90  b  +90 angular distance from galactic equator
towards N or S galactic poles.
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
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
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
. 4, 28 55)
(α,δ)  (17 42 m
Galactic rotation direction is (l, b)  (90, 0)
Prof. John Hearnshaw
h
Galactic coordinates: L is the galactic centre, G is the
north galactic pole, G' is the south galactic pole.
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
21. Spherical geometry
Spherical triangle
ABC is a spherical triangle (formed by 3 great circle arcs).
Prof. John Hearnshaw
This is a figure formed on the surface of a sphere by
three intersecting arcs of great circles.
ASTR211: COORDINATES AND TIME
Sides of the spherical triangle (a, b, c): measured in
degrees and equal to the angle between the radii from
sphere’s centre to two points of intersection of great circles.
Prof. John Hearnshaw
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.
The construction of spherical triangles from
great circles on the surface of a sphere
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
22. Spherical trigonometry
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
Prof. John Hearnshaw
Cosine formula:
cos a  cos b cos c + sin b sin c cos A
ASTR211: COORDINATES AND TIME
Note: for small a, b, c: cos a ~ (1  a2/2) etc.
Exercise:
Prove that a2  b2 + c2 2bc cos A
for a spherical  with small sides.
Prof. John Hearnshaw
The spherical  then approximates the plane .
ASTR211: COORDINATES AND TIME
Sine formula:
sin A sin B sin C


sin a sin b sin c
sin A sin B sin C


a
b
c
(sine rule for plane s).
Prof. John Hearnshaw
Note: for small a, b, c:
ASTR211: COORDINATES AND TIME
23. Applications of spherical trigonometry:
some practical examples
(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 ()
Prof. John Hearnshaw
a) Angle between two stars of given (, )
ASTR211: COORDINATES AND TIME
b) Relationship between (, ) and (a, A)
equatorial coordinates
Prof. John Hearnshaw
(H, ) = (hour angle, dec)
or (, )  (R.A., dec)
(a, A) = alt, az
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)
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
Also
cos z  cos (90  ) cos (90  )
+ sin (90  ) sin (90  ) cos H
sin a  sin  sin  + cos  cos  cos H
(2)
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
And
sin H sin(360  A)

sin z
sin(90   )
(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.
Prof. John Hearnshaw
sin H
sin A


cos a
cos 
ASTR211: COORDINATES AND TIME
Thus if (H, ) are known
obtain sin a from (2), hence a;
then get sin A from (3), hence A.
Prof. John Hearnshaw
If (a, A) are known
obtain sin  from (1), hence ;
obtain sin H from (3), hence H.
ASTR211: COORDINATES AND TIME
Azimuth, A, of a rising or setting object (z = 90º).
Prof. John Hearnshaw
c) Azimuth of setting/rising object of declination 
cos (90  )  cos (90  ) cos 90
+ sin (90  ) sin 90 cos (360  A)
 sin   cos  cos A
 cos A  sin δ / cos 
(or cos A   sin δ / cos  in S hemisphere)
Prof. John Hearnshaw
ASTR211: COORDINATES AND TIME
ASTR211: COORDINATES AND TIME
e.g. azimuth limits of setting Sun in Christchurch
  43 31; cos   +0.725
   23 27 (mid summer); = +23 27 (mid winter)
b) mid-winter: cos A  0.549
A  123.3 sunrise
or 236.7 sunset
Prof. John Hearnshaw
a) mid-summer: cos A  + 0.549
A  56.7 sunrise
or 303.3 sunset
ASTR211: COORDINATES AND TIME
A  90  33.3 at sunrise
 270  33.3 at sunset.
where the variation is the overall seasonal range.
Prof. John Hearnshaw
In general for Sun on horizon in Christchurch
ASTR211: COORDINATES AND TIME
d) Time of sunrise and sunset
This gives H, the hour angle of rising/setting Sun,
which is approximately the time interval between noon
and sunrise/set.
Prof. John Hearnshaw
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)
ASTR211: COORDINATES AND TIME
Example: length of daylight hours in Christchurch
  43 31; tan   0.950
mid-winter: cos H  +0.412
H  65.68  4.38 h
length of day  2H  8.76 h  8 h 45 m
Prof. John Hearnshaw
mid-summer: cos H  0.412
H  114.3  7.62 h
length of day  2H  15.24 h  15 h 15 m
ASTR211: COORDINATES AND TIME
e) Equinoctial corollaries
Prof. John Hearnshaw
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 observer’s latitude).
ASTR211: COORDINATES AND TIME
Also (from 23d)
cos H  tan  tan 
Length of day at equinox  2H  12 h
 length of night
(equinox  equal day and night).
Once again, this result is independent of , the
observer’s latitude.
Prof. John Hearnshaw
At equinox, tan   0
 cos H  0 or
H  90, 270
 6 h, 18 h (  6 h) (as 1 h  15)
ASTR211: COORDINATES AND TIME
Prof. John Hearnshaw
End of sections 18 - 23