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Table 826. Approximate Equation of Time

Table 826. Approximate Equation of Time

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T A B L E 827.-MlSCELLANEOUS



ASTRONOMICAL DATA *



729



A b e r r a t i o n constant.-20!'47

(conventional value ; work of Doolittle, Spencer Jones,

and others, indicates a value of 20'.'50).

Aphelion.-Point

where earth is farthest from sun = 1.520 X 10" cm.

Astronomical unit (A. U.)-Distance : mean distance earth to sun, 149,500,000 km.

(Conventional value, solar parallax 8!'79 would give 149,700,000.) Mass : the combined

mass of the sun and earth which means, practically, the sun's mass = 1.987 X 10" g.

Color index.-Ordinary

stellar magnitudes are supposed to correspond to observations

with the normal eye. This is by no means easy to define, for the brightness of a red star

compared with a white, appears greater when the amount of light entering the eye is

increased for both in the same ratio (Purkinje effect) for low brightness.

Owing to differences in the actual distribution of the energy with wavelength, the relative brightness of stars of different temperatures and colors measured with receptors

sensitive to different spectral regions vary greatly.

On ordinary photographs, red stars appear much fainter than to the eye. If the measures

are calibrated so that the visual and photographic magnitudes average the same for spectral class A , the difference for any other group of stars is called color index. This ranges

from about -0"'.3 to 1.8 for class &f and reaches 5" for the reddest stars of class N .

The difference in color index between the two standard types, e.g., A 0 and K O is

called the color-equation. It varies over a wide range with the spectral sensitivity of the

receiver, very large and positive for the violet and ultraviolet and negative for the red and

infrared.

Photoelectric devices, combined with screens and measurable transmission have at last

provided standard systems for stellar photometry of at least approximately definite physical significance for spectral regions ranging from the ultraviolet t o the infrared. Kndiometric magnitudes correspond to the measures of the whole observable energy radiition.

Bolometric magnitudes are supposed to represent the total energy radiation of all wavelengths, and must be found bv calculation.

Date line.-Established

by convention not far from the 180th meridian from Greenwich. Where the line runs across a group of islands, the change of the date line is diverted

to one side so that the group has the same day. Ships crossing from the east, skip a day ;

going east, count the same day twice.

Dav.-Mean

solar day = 1.440 minutes = 86.400 seconds = 1.0027379 sidereal day.

Sidereal day (ordinary, two successive transits of vernal equinox, might be called equinoctial day) = 86,164.09054 mean solar seconds = 23 hr, 56 min. 4.09054 sec mean solar time.

Two successive t r a n s i t s of s a m e fixed star = 86,164.09967 mean solar seconds.

Declination.-If

6 = declination, t , hour angle measured west from meridian, h,

altitude, @, latitude and A , azimuth measured from S. point through W. Then



+



++



sin h = sin @ sin 6 cos @ cos 6 cos t

cos h cos A = - cos @ sin 6 sin @ cos 6 cos f given 6, t, @

cos 6 sin t

cos h sin A =

sin6= sin @ sin h -cos @ cos h cos A

cos 6 cos t = cos @ sin h sin cos h cos A given h, A, @

cos 6 sin t =

cos h sin A



+



}

}



Delaunay's y = sin 1/2 I = 0.04488716 (Brown).

Dip of horizon.-In

minutes of arc = V elevation in ft (anproxivately).

r = 6.3712XlO' cm. Equatorial diameter = 12,756.78km ; polar diameter

Earth.-Mean

= 12,713.82 km. Area = 5.101 x 10" cm2. Angular velocity = 72.9X 10" radians/sec. Volume

= 1.083X10" cm8. Mass = 5.975X1On g. Density = 5.517 g/cm8. Mean distance to sun =

1.495)<10"cm. Distance to the moon = 3.844XlO'O cm. Light traverses mean radius of

earth's orbit in 498.6 sec. Semimajor axis orbit = 1.4950X1018cm ; semi ninor axis =

1.4948X10'8 cm. Viscosity = 10.9X10'' cgs. Velocity of equatorial point on earth, because

of rotation: 1,050 mi/hr = 1,550 ft/sec = 1,650 km/hr = 460 m/sec. I n orbit : 18/5 mi/sec

= 30 km/sec. See Tables 831 and 833. Rotational energy =2.16X10merg.

Earth's orbital velocity = 18.5 miles/second. 1,550 ft/sec (rotation at Equator).

E c c e n t r i c i t y of earth's o r b i t = e = 0.01675104 - 4.180X10-' ( t - 1900) - 1.26)<10-"

( t - 1900)'.

E c c e n t r i c i t y of moon's o r b i t = el = 0.05490056 (Brown).

Gal.-Unit

of gravity acceleration = 1 cm sec-'.

General precession (westward movement of the equinoxes) = 5072564 07000222

( t - 1900) per year (Newcomb). Probably requires correction of about -{- 0701. See

Table 838.

Gravitation constant = (6.670 2 0.005) X lo-* dyne cmzg-* (Heyl, 1930).

Gravity, acceleration due to, g = 978.0495 cm sec-' (conventional value at sea level at

equator. See Table 802). Unit, gal = 1 cm sec-'.



+



.



Prepared by G. M. Clemence, U. S. Naval Observatory.



(continued)

SMITHSONIAN PHYSICAL TABLES



730

T A B L E 827.-MISCELLANEOUS



ASTRONOMICAL DATA (continued)



H e a t index.-Radiometric

(heat or bolometric), zero taken to agree with Class AO,

(radiometric - visual magnitude) = heat index, for red stars.

Horizon.-Distance

at sea is approximately, miles = d (3/2) height in feet. Local

refraction (mirage) may introduce large percentage changes in either direction for observations from altitudes of 30 feet or less.

Inclination of moon's o r b i t = I = 5O843.5" (Brown).

Julian period, 1950= 6663.-January 1, 1950, Julian-day number = 2433283.

L a t i t u d e variation.-The

direction of the axis of the earth in space changes approximately 20'.'5 per year owing to precession. The change is roughly periodic in 25,800 years

with an amplitude of 23". This does not affect terrestrial latitudes, but a variation in

them is caused by a shift of the earth's body about this axis. The two ascertained components of the polar motion have periods of 1.00 and nearly 1.20 years (the annual and

Chandlerian components, respectively), so that the oscillations in X and Y, as well as the

resultant total motion have variations in amplitude with a "beat period" of about 6 years.

In contrast to the annual terms, Chandler's term shows striking variations in am litude.

There is, further, a variation in the period of the Chandlerian term (1.18, 1.20, 1.17, 1.15,

1.19 years) which appears nearly proportional to the corresponding amplitude variations

according to the relation P = 0.185 A +1.128, where P is the period in years and A the

amplitude in 0'.'01 units. (See T. Nicolini, appendix to Commission 19 Report, Trans. Int.

Astron. Union, Zurich, 1948.)

Light, velocity of.-(Mean value) in vacuo, 299773 rt 10 km sec-' (Dorsey).

299792.5 k 0.8 km sec-' (Bearden).

299776 0.00004 km sec-' (Birge).

Light year.-The

distance light travels in 1 year = 9.5 X 10" kilometers = 5.9 X 10"

miles. Light traverses mean radius of earth's orbit in 498.6 sec.

L u n a r inequality of earth = L = 6.454:'

L u n a r n o d e d = daily motion = - O"52954.

L u n a r parallax = 3422.70" (Brown).

L u n a r perigee, daily motion = 0?111404.

Lunar-solar precession = p' = 50.3714" per year (De Sitter, 1927). Of this 0.0191",

Einstein, orbital motion earth.

Magnitudes.-The

observed intensity of light received on the earth from astronomical

bodies ranges over a factor exceeding 10". I t is therefore expressed on a logarithmic

scale-the system of stellar magnitudes. This system, which was adopted by Hipparchus

more than 2,000 years ago, is closely represented by the equation

m = 2.5 loglo ( l o / l )

where 1 is the observed light and 10 a standard value corresponding roughly to the light

of Arcturus or Vega. Decrease of light by a factor of 100 increases the stellar magnitude

by 5.00 ; hence the brightest objects have negative magnitudes. (Sun : - 26.8 ; mean full

moon : - 12.5 ; Venus at brightest : - 4.3 ; Jupiter at opposition : - 2.3 ; Sirius : - 1.6 ;

2.1). The faintest stars visible to the naked eye on a clear dark

Vega :

0.2 ; Polaris:

night are of about the sixth magnitude (though on a perfectly black background the limit

for a single luminous point approaches the eighth magnitude). The faintest stars visible

with a telescope of aperture A (in inches) is one approximately of magnitude 9 5

log,o A. The magnitude of the faintest stars which can be photographed with the 200-inch

22.7. The apparent magnitude of a standard candle at a distance of

telescope is about

1 meter is - 14.2.

Absolute macnitude, M, is that which the body would exhibit if placed at a distance of

10 parsecs, and corresponds to its actual luminosity. For a star of magnitude m, and

parallax p , in seconds of arc

M = m 5 5 log$

For the sun, M = 4.7. The brightest stars probably exceed M = - 7 and the faintest

observed value is M = 18, a range of 10". The full moon (could it be observed without

interference from the standard distance) would have M = 32 and a standard candle

72.8.

M e a n distance earth t o moon = 60.2678 terrestrial radii.

= 384,411 kilometers = 238,862 miles. (See Table 834.)

M e a n distance earth to sun = 149,500,000 kilometers = 92,900.000 miles. (See Astronomical unit.) See Table 833.

Month.-Sidereal

= 27.321661 days, synodical (ordinary) = 29.530588 days (Brown).

Nutation constant (periodic motion of celestial pole) = 9.21". conventional value ;

9.207:' Principal in long = A 6 = (-17.234" - .017" T) sin 0 ; principal term in obliquity = A o = (+9.210 . O W T) cos 0 (Newcomb). T centuries from 1900.

Obliquity of ecliptic = 23"27'8.26" - 0.4684 ( t - 1900)" (Newcomb).



+



*



+



+



+



+



+



+



+ +



+



+



+



+



(continwd)

SMITHSONIAN PHYSICAL TABLES



73 1

T A B L E 827.-MISCELLANEOUS



ASTRONOMICAL DATA (concluded)



Parallactic inequality m o o n = Q = 124.785" (Brown.)

of star whose parallax is 1 sec = 31 X lo'* km = 19.2 X 10" miles

Parsec.-Distance

= 3.263 light years.

Perihelion.-Point

where earth is nearest sun = 1.4700 X lo" cm.

P l a n e t a r y precession = X = 0.1247" (Newcomb).

Pole of M i l k y W a y = R. A., 12 hr 48 min; Dec., 27:

t"F)I tan 2,where 2 =

Refraction.+

in. (") = [983 X (barometer in in.)/(460

zenith distance. Error < l", 2 < 75", ordinary t and pressure.

Solar diameter = 864,408 miles.

Solar parallax = 8230 (conventional value), 8Y79 (Newcomb, Spencer Jones).

Sun.--r = 6.965 x 10" cm. Area = 6.093 X lomcm'. Volume = 1.412 X 10" cm'. Mass

= 1.987 x 10mg. Density = 1.41 g/cmS. Mean distance to earth 1.495 X 10"cm. See

Table 831.

Twilight.-There

are three definitions of twilight : civil, nautical, and astronomical.

Civil twilight lasts until the sun is about 6" below the horizon, after which motor-car

lights must be turned on. Nautical twilight lasts until the sun is about 12" below the

horizon. This is the limit for observations of stars with the sea horizon. Astronomical

twilight is considered to end when the sky is dark in the zenith. It lasts until the sun is

about 18" below the horizon. For latitudes > 50" there is a faint twilight at midnight

in midsummer.

Year.-Anomalistic

(two successive passages of the perihelion) = 365.25964134 3.04

x lo-' ( t - 1900) days. Eclipse (time taken by sun to pass from a node of the moon's

orbit to the same node) = 346.620031 3.2 X lo-' ( t - 1900) days. Sidereal (from given

star to same star again) = 365.25636042 + 1.1 x lo-" ( t - 1900) days. Tropical (ordinary)

(two successive passages of vernal equinox by sun) = 365.24219879 - 6.14 X lo-'

( t - 1900) days.



+



+



+



+



T A B L E 828.-ELEMENTS



O F SOLAR MOTION



*



Because of the asymmetry in stellar motions (Table 876), determinations of the speed

and direction of the sun's motion are very sensitive to the selection of stars to which it is

referred. Ideally we wish to refer the sun's motion to the circular velocity with respect to

the galactic center; this may be called the basic solar motion. It is possible to determine

this basic solar motion from detailed studies of the distribution of motions among nearby

stars and it is found that such a determination made from the giant K stars is in excellent agreement with an independent determination from the A stars (Janssen and Vyssotsky). This value is given in the last line of the table. The figures listed for the first five

groups are smoothed values obtained from a combination of the best observational results.m The values for the next four groups come from investigations made at Leiden,

Mount Wilson, an& McCormick Observatories. The solar motion with respect to B stars,

c-stars, and Cepheids is difficult to determine satisfactorily because of uneven distribution in space, very small proper motions, etc.

Coordinates of the apex

Stellar group

of reference



B8 to A3. .................

A5 to F2 ..................

F5 to GO ..................

KO to K 2 . . ................

gK5 to gM8 ...............

dK8 to dM5 ...............

Irregular var ..............

Long-period var ...........

Cluster-type var ..........

Basic solar motion .........



Solar

velocity



16 km/sec

17

18

20

22

23



35

54

130

15



RA



263"

266

269

273

276

275

265 :

295

297

260



Prepared by A. N. Vyssotsky, University of Virginia.

Astron. Journ., vol. 53. p. 87, 1948.



SMITHSONIAN PHYSICAL TABLES



Dec



+zoo

+23

+26

+29

+31

+44

+38 :

+46

+52

+17



Gal

long



11

0

15



Gal

lat



+24"

22

18

+21

23

+19

27

+17

39

22

+28 :

30 :

47

+10

53

12

+25

7

-~~



+



+

+



732



T A B L E 829.-PERPETUAL



CALENDAR *



This calendar gives the day of the week for any known date from the beginning of the Christian

Era down to the year 2400.

Dominical letters

Century

Year



Julian Calendar

100

800

1500t



200

900



300

1000



400

1100



500

1200



600

1300



DC

B

A

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA

G

F

E

DC



ED

C

B

A

GF

E

D

C

BA

G

F

E

DC

B

A

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED



FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA

G

F

E

DC

B

A

G

FE



GF

E

D

C

BA

G

F

E

DC

B

A

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF



AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA

G

F

E

DC

B

A

G



BA

G

F

E

DC

B

A

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA



CB

A

G

F

ED

C

B

A

GF

E

D

C

BA

G

F

E

DC

B~

A

G

FE

D

C

B

AG

F

E

D

CB



0



1 29 57 85



2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28



30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56



58 86

59 87

60 88

61 89

62 90

63 91

64 92

65 93

66 94

67 95

68 96

69 97

70 98

71 99

72

73

74

75

76

77

78

79

80

81

82



83

84



FE



D

C

B

AG



1500$ 1600

2000



-_

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C



-_

-_

-_



1700

2100



BA

G

F

E

DC

B

A

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA



C

B

A

G



FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF

E

D

C

BA

G

F

E

DC



1800

2200



E

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

A

GF



E

D

C

BA

G

F

E

DC

B

A

G

FE



1900

2300



G



F

E



D

CB

A

G

F

ED

C

B

A

GE

E

D

C

BA

G

F

E

DC

B

A

G

FE

D

C

B

AG



Dominical letter



Month



June

Aug.

Sept., Dec.

1 8 15

2 9 16

3 10 17

4 11 18

5 12 19

6 13 20

7 14 21



Gregorian Calendar



0

700

1400



22. 29

23 30

24 31

25

26

27

28



F



A

D

G

B

E

C



B

E

A

C

F

D

G



C

F

B

D

G

E

A



D

G

C

E

A

F

B



Mon.

Tues.

Wed.

Thurs.

Fri.

Sat.



Sat.

Sun.

Mon.

Tues.

Wed.

Thurs.

Fri.



Fri.

Sat.

Sun.

Mon.

Tues.

Wed.

Thurs.



Thurs.

Fri.

Sat.

Sun.

Mon.

Tues.

Wed.



F

Sun.



G

C

F

A

D

B

E



B

E

G

C

A

D

Wed.

Thurs.

Fri.

Sat.

Sun.

Mon.

Tues.



Tues.

~.

Wed.

Thurs.

Fri.

Sat.

Sun.

Mon.

~



Mon.

Tues.

Wed.

Thurs.

Fri.

Sat.

Sun.



T o find the calendar for any year of the Christian Era, first find the Dominical letter for the

pear in the upper section of the table. Two letters are given for leap years ; the first is to be used

for January and February, the second for the other months. I n the lower section of the table,

find the column in which the Dominical letter for the year is in the same line with the month

f o r which the calendar is desired ; this column gives the days of the week that are to be used with

the month.

E.g., in the table of Dominical Letters ,ve find that the letter for 1951 is G ; in the line with

July, this letter occurs in the first column; hence July 4, 1951, is Wednesday.

Prepared by G . hf. Clemence, U. S. Naval Observatory.

and after 1582, Oct. 15 only.



SMITHSONIAN PHYSICAL TABLES



t On



and before 1582, Oct. 4 only.



$On



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