## Sample Exercises and Problems

Compute the maximum altitude achieved by the Sun during the year at sites with latitude = 90°, 66.7°, and 231/2°.

Calculate the maximum azimuth of the setting summer solstice Sun at sites with latitude = 51° and 32°. Calculate the maximum elongation of (a) Mercury and (b) Venus as seen from the Earth; use mean distances from the Sun for all planets.

Calculate the hour angle and declination for an object at azimuth 120° and altitude 50° at a site with latitude = 45°.

Do the calculation in Question 4 for a site with latitude -45°, and comment on the required convention for treating Southern Hemisphere site calculations. Compute the right ascension and declination for an object with celestial longitude 75° and celestial latitude 5°.

Calculate the arc distance between two objects on the sky separated by 45° of right ascension and 20° of declination.

At a certain observatory, Orion's Belt is observed to rise parallel to the horizon. From star charts and spherical astronomy, determine the latitude of the site. Calculate the difference (a) in azimuth and (b) the difference in hour angle for two objects on the ecliptic, one at 0° and the other at +10°.

At a certain site a 1- or 2-day-old crescent Moon is observed to have its horns pointing up, directly away from the horizon. If the date is Sept. 21, what is the latitude of the site?

(2) Compute the refraction and the observed altitude under standard atmospheric pressure and temperature conditions for objects at the following altitudes [see §3.1.3 and (3.16)]: h = 30°, h = 45°, h = 60°, and h = 90°.

(3) Calculate the expected (algebraically) maximum altitude of the Sun at midwinter from Novaya Zemlya [see §3.1.3 and (3.16)].

(4) Calculate the effective magnitude of a cluster of 50,000 stars of average magnitude 16 [see §3.1.2.4.5 and reasoning behind (3.13)].

(5) Calculate the apparent azimuth of sunrise at midsummer at a site with latitude = 51°. Assume dz = 1/2° [see §3.1.3 and (2.1)].

(6) Taking refraction into account, what is the algebraically smallest latitude at which the phenomenon of the "midnight Sun" can be observed?

(7) From the precessional pole charts (Figures 3.9 and 3.10), which of the first magnitude stars were circumpolar (a) at Giza (Cairo will do!) at 2500 b.c. and (b) at Callanish, 1500 b.c.?

(1) Calculate the hour angles of the onset of civil and astronomical evening twilight at a) the equator and b) a latitude of 51°N. [See §3.1.2.5 and (2.5).]

Calculate the angle between the shadows' edges of a vertical gnomon cast by the Sun at 12 and 1 p.m. local solar time for a flat sundial.

Calculate the length of the solar shadow at noon at a site with latitude 45° for such a sundial at (a) summer solstice and at (b) winter solstice. Calculate the length of daylight for Alexandria in winter and sunlight, correcting for expected mean refraction and the semidiameter of the Sun.

Determine the length of astronomical twilight for Alexandria (as per Question 3).

Compute the altitude of Thuban (alpha Draconis) at 2500 b.c. at Giza (Cairo will do!) at upper culmination.

(6) Derive the mean length of the synodic month from that of the mean sidereal month length and the sidereal year length.

(7) Derive the length of the tropical year from the sidereal year and the precession rate.

(8) Calculate the length of a "seasonal hour" at winter solstice at (a) Rome and at (b) Karnak.

(9) Calculate the length of a "seasonal hour" at summer solstice at (a) Rome and at (b) Karnak.

(1) Determine the maximum altitude of a sundog for a setting/rising Sun.

(2) Compute the maximum altitude of the primary and secondary rainbows of the rising/setting Sun.

(3) Compare the azimuths of the rainbows of Question 2 for a setting summer solstice Sun at latitudes = 20° and 60°.

(4) Calculate the brightness required for the Crab Nebula supernova to be visible in the daytime.

(5) Estimate the energy produced by the impact of a 50-km-diameter comet on the forward (Eastern) limb of the Moon. Assume maximum possible velocity of impact.

(6) Estimate the brightness of the impact described in Question 5 for an observer on Earth. Explicitly list and discuss all assumptions.

(7) Discuss the arcus visionis needed to see a first magnitude star when it is (a) 1° and (b) 10° above the astronomical horizon above the Sun.

(1) Demonstrate the correctness of the statement in fn. 4 of §6 concerning the identical equations derived for the Southern Hemipshere.

(2) Calculate the amplitudes of the rising/setting Sun at (a) Stonehenge and at (b) Tenochtitlan for appropriate epochs.

(3) Calculate the extreme amplitudes of the rising/setting Moon at (a) Stonehenge and at (b) Callanish.

(4) Compute (a) the Julian day number and (b) the back-calculated Gregorian date of October 31, 1517.

(5) Determine the date of Easter for the current year (stating the criteria and whose criteria they are).

(6) Discuss the importance and limitations of the probability approach to deciding the "reality" of astronomical alignments.

(7) A commonly discussed problem is how the pyramids could have been aligned as accurately as they apparently are. List and discuss several astronomically based schemes for doing so.

(8) Compute by interpolation (from the data given in Table 2.3) the expected lengths of the seasons for the epoch of the construction of Angkor Wat (§§9.3 and 15.3.2). Can you think of an alternative interpretation for the numbers of asuras and devas?

(9) Suppose you have an eroded Maya monument of which you can read:

Give the correct reading of the Long Count and the Calendar Round.

(10) Suppose you infer from a myth that Jupiter, Saturn, and Venus were "close together" in the sky in the constellation Gemini. Define "close together," and calculate the approximate dates when this would have been true in the past 3000 years.

(11) Suppose a painting in a cave depicts a deity whom you have identified as Saturn and is illuminated at the winter solstice. What inferences would you consider legitimate as to the astronomical conditions when the painting was created? About how often would those conditions repeat?

(12) On what days of the retrodicted Gregorian year would zenith passage of the Sun occur at latitude 21°15'N?

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