Astronomy 111 | Recitation 1

Astronomy 111 -- Recitation 1

Prof. Douglass 5?6 September 2019

Formulas to remember

Blackbody radiation Stefan's law: f = T 4, where = 5.67051 ? 10-5 erg s-1 cm-2 K-4

Planck function:

2hc2

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u = u(, T ) = 5 ehc/kT - 1

where h = 6.626 ? 10-27 erg s and k = 1.381 ? 10-16 erg K-1. The dimensions of the Planck func-

tion are power per unit area per unit solid angle per unit wavelength interval.

Wien's law: maxT = constant = 0.2897756 cm K

Cartesian, spherical, and celestial coordinates

x = r sin cos y = r sin sin z = r cos

r = x2 + y2 + z2

= tan-1

x2 +y 2 z

= tan-1

y x

If the x-axis points toward the Vernal equinox and the z-axis toward the north celestial pole,

then

= = 90 -

(1)

Figure 1: Celestial (left), spherical (middle), and Cartesian (right) coordinate system.

Workshop problems

Warning! The workshop problems you will do in groups in Recitation are a crucial part of the process of building up your command of the concepts important in AST 111 and subsequent courses. Do not, therefore, do your work on scratch paper and discard it. Better for each of you to keep your own account of each problem, in some sort of bound notebook.

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1. Distances on the surface of a sphere (a) The radius of the Earth is R = 6.4 ? 108 cm. How long (in cm) is the equator? How long (in cm) is the 45 parallel? (Hint: What is the radius of the 45 parallel?) What is the distance (in cm) between two points on the equator that lie 0.02 radians of longitude apart? Between two points at latitude 45? Both calculate these quantities and illustrate them with a drawing. (b) Generalize the previous result: what is the distance, d, between two points at latitude , that differ in longitude by L? (c) Adapt the previous result: in the view of an observer at the Earth's center, what is the angle (in radians) between two points on the Earth's surface at latitude that differ in longitude by a small amount L? (d) And generalize that result. The celestial coordinate system which astronomers use to describe the location of objects on the sky is the projection of the lines of latitude and longitude onto the celestial sphere. The angle corresponding to longitude is called right ascension ("RA," usually given the symbol ) and declination ("Dec," or ). The zero point of RA is one of the two places at which the celestial equator (the projection of the Earth's equator on the celestial sphere) intersects the plane of the Solar System (the zodiac), and increases toward the east. See Figure 2.

Figure 2: The celestial coordinate system. The intersections between the celestial equator and the ecliptic (a.k.a. zodiac) are called the equinoctes. One of these, the vernal equinox, is the zero point for right ascension. The vernal equinox is where the Sun is on the first day of Spring.

So, in the view of an observer on Earth, what is the angle (in radians) between two points at declination on the celestial sphere that differ in right ascension by the small amount ? 2. Blackbody radiation (a) Estimate your body's surface area A (in cm2), your skin temperature (in K, assumed uniform over your body), and the room's temperature (also in K).

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(b) How much power, Pem (in erg/s), does your body emit in blackbody radiation? (c) How much power, Pabs (in erg/s), does your body absorb in the room's blackbody radia-

tion? (d) Suppose the balance between these two powers has to be made up by chemical energy

released in your body, ultimately from the digestion of food. How much power, Pfood (in erg/s), does your body need to produce? How many calories per hour is that equal to? (Note: a dietary calorie, which is what we are referring to here, is what chemists call a kilocalorie and is equal to 4.18 ? 1010 erg.)

Learn your way around the sky, lesson 1. (An exclusive feature of AST 111 recitations.) Use the lab's celestial globes, Stellarium (free software available for download), SIMBAD at http: //simbad.harvard.edu/simbad/sim-fid, and any other resources you would like to use, to answer these questions about the celestial sphere, the constellations, and paths of the planets through the sky.

6. You are in Rochester staring at a field of stars that lies on the celestial equator and envisioning their celestial coordinates. In what direction do the stars' right ascensions increase, as one moves across the starfield? Check this by consulting the celestial globes in the cabinet.

7. Normally, astronomers report right ascension in hours instead of degrees because the rotation of the Earth makes the sky march across your view so that the whole sky is covered in one rotation. How many degrees are there in one hour of right ascension?

8. In a week or two you will derive (here in recitation) this approximate formula for the angle between two celestial coordinates, (1, 1) and (2, 2):

= (1 - 2)2 + (1 - 2)2 cos 1 cos 2 1/2

Use this to calculate the angular length of the base of the Big Dipper, defined by the stars Merak and Phecda. Look up their coordinates in SIMBAD.

9. Discussion question. You are Christopher Columbus. You want to get rich by finding a short cut to the Indies, China, and Japan. All you know about those places is that Japan is supposed to be at a latitude similar to Spain's, that China is much bigger and stretches south from Japan, and the Indies are a collection of large and small islands east of a jungle-covered peninsula running NW?SE (the Malay peninsula) that stretches down from China. You have seen many estimates of the size of Asia and the circumference of the Earth and have noted that if one takes the largest estimate for Asia and the smallest for Earth, that these places could be as close to Spain as 3000 miles. You are also a good sailor and know well the pattern of winds in the eastern Atlantic. So, you set off on your journey. Here is what you see:

(a) There are indeed islands -- large ones mixed with small ones -- only a few thousand miles due west of the Canary and Cape Verde Islands (that is, well south of Spain). Remember that distances E?W were difficult to measure in Columbus' day.

(b) These islands seem to bound a mirror image of the winds in the east: a circulation pattern whose N?S extent is a few thousand miles -- consistent with the dead-reckoning estimate of the E?W distance.

(c) Traveling further west, one finds an unbroken, jungle-covered coastline running roughly NW?SE.

(d) The islands and the unbroken coast are inhabited by people who resemble the oriental people as described in Marco Polo's book.

(e) Further south -- past all the islands -- another unbroken coast is seen, from which drain rivers sufficiently large that this land must be a continent, not an island.

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Columbus concluded from these observations that he was in the Indies. Now we know that he was not, of course, but thinking as a scientist: what was wrong with his reasoning, and what clues should he have interpreted differently that would have put him on the right track? 10. Observation planning. When is the next night that is likely to be clear at Mees Observatory? Try or type=text&Mn=telescope. 11. Celestial, spherical, and Cartesian coordinates. A star has right ascension and declination and . (a) Draw a Cartesian coordinate system with the x-axis pointing through = = 0 and the

z-axis pointing toward = 90. Write an expression for a unit vector pointing at the star in terms of the Cartesian unit vectors x^, y^, and z^. (b) Recall that the inner ("dot") product of two vectors A and B can be written in these two ways:

A ? B = AB cos = AxBx + AyBy + AzBz

where A and B are the magnitudes of the two vectors, is the angle between them, and the subscripts x - y - z indicate the components of the vectors along those Cartesian axes. Two stars have celestial coordinates 1, 1 and 2, 2. Use your result from part a to obtain an exact expression for the angle between (the directions toward) these stars. Remember this resulting equation: it is one of the two equations that are fundamental for understanding angular distances along the celestial sphere. (Hint: for simplification, recall the trigonometric identity cos(a ? b) = cos a cos b sin a sin b.)

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