3.2 Coordinates on a celestial sphere

Problem 3.21. (a) What are the approximate equatorial coordinates of the north celestial pole, of the sun on March 21, the star shown in Fig. 1, and the nominal galactic center at l = b = 0? In each case, specify the epoch of your answer. What are the J2000 coordinates of the Crab nebula? (b) Locate Norton's 2000.0 Star Atlas in the library and find the J2000.0 coordinates (to the nearest arcminute) of the variable and bright star 7 Cassiopeiae, the Seyfert galaxy NGC 1068, the variable star ^ Carinae, and the globular cluster M 30? You can measure from the charts or use the "2000" coordinates listed in the several tables of the Atlas; the Atlas suppresses the "J" prefix. [Ans. (a) galactic center a « 18 h, 8 « -30°]

Problem 3.22. (a) Plot the following objects on a copy of Fig. 5 and determine by inspection the galactic coordinates of the objects as best you can (e.g., to about 20° precision, though in some instances one can do better): the Crab nebula at a (B1950) = 5 h 32 m, 8 (B1950) = +21° 59', the radio galaxy M87 (12 h 28 m, + 12° 40'), the nearby Andromeda galaxy M31 (+0 h 40 m, +41° 00'), and the central star in Cassiopeia called Gamma Cas = 7 Cas (00 h 54 m, +60° 27'). (b) Program your calculator (or the equivalent) with Eqs. (6)-(10) to obtain accurate values (to about 0.1° precision) of the galactic coordinates of the above objects. Pay attention to the signs of your trig functions. (c) If an astronomical plastic globe with celestial equatorial coordinates indicated on it is available to you, determine the above answers by making measurements (with a piece of string) along the surface of the globe. Your answers should be good to about 2 or 3 degrees.

Problem 3.23. (a) Make a sketch of a celestial sphere with the ecliptic as the equator. Indicate the path followed by the sun. Show the track scanned by the view direction of an x-ray detector on a satellite in space if its view direction is normal to the sun direction and it rotates once per hour about the axis directed toward the sun. The spin axis remains pointed toward the sun as the sun moves along the ecliptic. The field of view (FOV, full width) is circular with diameter 2°. (b) What happens to the scan track as the year progresses? Approximately how many times a year is a celestial source transited by at least some part of the FOV if the source lies on the ecliptic equator, or if it is at ecliptic latitude +80°? For some transits, the view will be blocked by the earth; assume this never happens. [Ans. (b) ~ 100, -600]

3.3 Solid angle on the celestial sphere

Problem 3.31. (a) Derive an expression for the solid angle of one polar cap on the celestial sphere, of angular radius 9. (On the earth, the polar cap could be encompassed by the arctic circle.) What is the fraction of the entire sky that is contained in this polar cap of radius 9 ? What is this fraction if 9 = 30°? (b) In the limit of small 9, i.e., 9 ^ 1, show that the solid angle is simply the geometric flat-space "area" ^92 measured in square radians. (c) Use this result to recalculate the solid angle of the 30° polar cap. By what fraction is it in error? (d) Repeat (a) and (c) for 9 = 1° and for 9 = 60°. [Ans. -7%; —; -2%; —10-3 sr and 0.003%]

Problem 3.32. (a) What is the approximate solid angle of the sun as viewed from the earth, in square arcmin?, square degrees?, steradians? The sun's angular diameter is 32'. (You may use the geometric flat-space approximation.) (b) Estimate the solid angle (in steradians and sq. arcsec) of the sun-like star a Centauri of radius —7 x 108 m, which is at a distance of 4.4 LY. [Ans. —10-4 sr; —10-15 sr]

3.4 Surveys, charts and catalogs

Problem 3.41. Compare the problems you would encounter in trying to keep track of the positions of all the trucks in the United States as time progresses with the problems facing astronomers in keeping track of all stars as time progresses.

Problem 3.42. Use Norton's 2000.0 Star Atlas as a finding chart for the bright naked-eye objects. Locate a constellation of interest to you, e.g., Orion, on the star chart and pick out a faint 5th or 6th magnitude star near one of the bright stars on the sky map. Try to find it with your naked eye or with binoculars. Do this on a dark night (moon set) away from city lights if possible, and be sure to let your eyes become dark adapted. (Use a faint red-light flashlight when examining the catalog.) Find the Orion nebula, labeled 42M for Messier 42, in the sword of Orion. Can you see its nebulosity with binoculars? Mount your 35-mm camera on a tripod and take several time exposures with the focus at infinity and the aperture wide open. Try exposures: 1 s, 10 s, 60 s, 90 s, 2 min, 3 min, 5 min, 10 min. Can you see more or fewer stars in the photos than with binoculars? Do the lengths of the tracks in the prints match the rotation rate of the earth?

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