Space, Time, Matter and Cosmology
The geodetic, or geographic, surface position is defined by a grid of great circles on a spherical Earth. A great circle divides the sphere in half, and the name comes from the fact that no greater circles can be drawn on a sphere. A great circle halfway
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Symbol
Value
Hubble constant t
Ho
100 h km s- 1Mpc-1
Normalized Hubble constant t
h=Ho/100
0.50 < h < 0.85
Hubble timet
1/Ho
9.778 13 x 109 h- 1years
Age ofUniverset
to
15(5) x 109 years
Critical density of Universe t
p.= 3H~/8nG
1.878 82(24) x 10-29 h 2 g cm-3 2.775 366 27 x 1011 h 2 M 0 Mpc-3 1.053 94(13) x 10-5 h 2 GeV cm-3 0.1 - 0.0000058 sin2 2¢) .
4>
is
(5.24)
The position of a point on the Earth can also be given in relationship to the Earth's center by specifying the geocentric longitude, latitude, and distance, A., 4>', p, or the geocentric equatorial rectangular coordinates, x, y, z. The difference between the geodetic latitude,¢, and the geocentric latitude, 4>', is given by 4>- 4>' = (692.74 sin2¢- 1.16 sin 4¢) seconds of arc . (5.25) Conversion from geodetic (geographic) to geocentric coordinates is straight forward using the following relationships (Seidelmann, 1992):
= p cos 4>' cos A. = (aeC +h) cos 4> cos.A. , y = p cos 4>' sin A. = (aeC +h) cos 4> sin A. , z = p sin 4>' = (aeS + h) sin 4> x
(5.26)
where ae is the equatorial radius of the spheroid, and C and S are auxiliary functions that depend on the geodetic latitude and the flattening f of the reference spheroid. It follows from the properties of the ellipse that C
= (cos2 4> + (1- j) 2 sin2 ¢)-!,
= (1- f) 2 C
S
(5.27)
.
The geocentric distance, or radius, is often given in units of the equatorial radius of the reference spheroid, obtained by dividing the value given here by Series expansions, which contain terms up to f
s=
~f
l-
2
1+
~f 2
for S, C, p, and
4> - 4>' are
~ !2 + ~!3 - (~ f - ~ !2 - ~!3) cos 2¢
+
16
+ c 36/2
c=
3,
+
32
:2!3 )
-
2
2
cos4¢- :4f3 cos6¢ '
~ !2 + }___!3 - (~ f 16
+ c36/2 +
64
32
+
2
~!2 + 27 !3) cos 2¢ 2
64
:2!3) cos4¢- :4!3 cos6¢
3 ( 2!-64! 1 13 3) cos 2¢ 5 12+ 5 1 3) cos4¢ + 13 1 3cos6¢ , ( 16 1
5
2
(5.28)
'
5
p= 1 -2/+16/ +32/ +
-
4>- 4>' =
32
(1
+
~!2 ) sin 2¢-
64
G/
2+
~f3 ) sin4¢ + ~f3 sin 6¢
.
6
5. Space, Time, Matter and Cosmology
The expressions for p and o•·
320
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300
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.
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,
-
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eeL . '
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60
.. r~ / >.-· · ·-"'-i-.e •. ··;·
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120
,,.v· ·.~I
1-•" " ..... : ·. ·.f/ ... ~ 1" .. -" \ · . T 1< 1' k'r/ \ 1.:>.· • l>i\. ' r.., "':~'"" r:(- ~. · ·. '[lfli n, 1;··~ 71 .. . z::;~~J \ I>·~ \J': ; t;~;f·\ · ~- }dY'! I . ,.y b.,~J~: T' .... !{,;:I>< ·"'"" ')
1.0
L~;
ro~ 25~
~~,r,t.;._
30 -04
00
0.4
as
l2
LS
2"G
2A
28
8-V
Fig. 5.8. The two-color diagram for over 29,000 stars. The difference between the apparent magnitudes, m, are plotted for the ultraviolet, U, blue, B, and visual, V, spectral regions with U- B = mu- mo and B- V = mJJ- mv. (Courtesy of the Geneva
