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Can you recommend any books on the Hebrew Calendar or Jewish Holidays?

Jewish Holidays: A Guide & Commentary For a good reference book on the major Jewish Holidays, I’d suggest Michael Strassfeld’s The Jewish Holidays: A Guide & Commentary.

Rabbi Nathan Bushwick’s Understanding the Jewish Calendar explains some of the mathematical and scientific underpinnings of the calendar.

Calendrical Calculations For a comprehensive look at 25 different calendar systems (including Hebrew, Gregorian, Julian, Mayan and Hindu), and corresponding sample code in Lisp and Java, Reingold & Dershowitz’s Calendrical Calculations is an unparalleled resource.

Can I download a copy of the hebcal.com program for my computer?

Hebcal for UNIX is a free program and is available for download at https://github.com/hebcal/hebcal. This is the same engine that powers the hebcal.com website. The downloadable program has a text (command-line) interface, not a fancy graphical one.

Kaluach is an unrelated program available at http://www.kaluach.com/. The author requests donations, but does not have a set price.

When does Shabbat begin?

Shabbat begins 18 minutes before sundown on Friday night. In Jerusalem, Shabbat begins 40 minutes before sundown.

According to the United States Naval Observatory,

Sunrise or sunset is defined to occur when the geometric zenith distance of center of the Sun is 90.8333 degrees. That is, the center of the Sun is geometrically 50 arcminutes below a horizontal plane. For an observer at sea level with a level, unobstructed horizon, under average atmospheric conditions, the upper limb of the Sun will then appear to be tangent to the horizon. The 50-arcminute geometric depression of the Sun’s center used for the computations is obtained by adding the average apparent radius of the Sun (16 arcminutes) to the average amount of atmospheric refraction at the horizon (34 arcminutes). [USNO]

What is Havdalah (or, When does Shabbat end)?

Shabbat ends after sundown on Saturday night when there are three stars visible. Depending on latitude and longitude, this is usually between 42 and 72 minutes after sundown.

According to Wikipedia,

There are three widely observed practices, all of which have support in the halachic literature:

  • Appearance of three medium-sized stars in the sky (sun 7°5′ below the horizon, or 42 minutes after sundown), as in the Talmud. This is normative practice in Conservative Judaism. In Orthodox Judaism, this position is used widely for the end of rabbinical fasts, but less frequently for the end of Shabbat or biblical festivals.
  • Appearance of three small stars widely spaced in the sky (sun 8.5°-8.75° below the horizon): common practice in much of Orthodox Judaism [10]
    • “50 minutes after sundown” is actually a variant of this position. Rabbi Moshe Feinstein ruled this way because most people cannot easily calculate when “8.5° after sundown” will occur, and 8.5° takes 50 minutes at its longest, near the summer solstice, at the latitude of much of the United States.[2]
  • 72 minutes after sundown (“opinion of Rabbeinu Tam“): equivalent to other definitions of nightfall, and safe according to all opinions. Common practice in Chasidic and other Charedi communities

If I use 72 minutes for Havdalah, why are Havdalah times ~90 minutes later than the previous day’s candle-lighting time?

Shabbat begins 18 minutes before sundown on Friday night.

Havdalah is 72 minutes after sundown on Saturday night.

So in a given week, Havdalah time is typically 90 minutes after the previous day’s candle-lighting time (18 + 72 = 90). Sometimes there’s an extra minute or two difference, and that’s due to sunset actually differing by a minute or two between Friday and Saturday nights.