Qibla observation by shadows
Twice every year, the Sun culminates at the zenith of the Muslim holy site of Ka'bah at the local solar noon, allowing the Muslim sacred direction, or the qibla, to be found in other parts of the world by observing the shadows cast by vertical objects. These moments occur on 27 or 28 May, and on 15 or 16 July. During these times, the Sun appears in the direction of the Ka'bah when visible from any observer on earth, and its shadows cast by vertical objects indicate the same direction. On the two other moments, the Sun passes the nadir of Mecca, resulting in the exact opposite direction but can serve the same purpose for determining the qibla. These occur on 12, 13, or 14 January at 00:30 SAST and 28 or 29 November at 00:09 SAST.
This phenomenon occurs because the Sun path makes the subsolar point travel through every latitude between Tropic of Cancer and the Tropic of Capricorn every year, including the latitude of the Ka'bah, and because the Sun crosses the local meridian once a day. This was known since at least the thirteenth century by the astronomers Jaghmini and Nasir al-Din al-Tusi, but because the Islamic world used the lunar instead of the solar calendar, their timing could not be stated as a fixed date.
Context
Qibla
The qibla is the direction of the Ka'bah, a cube-like building at the centre of the Sacred Mosque in Mecca, in the Hejaz region of Saudi Arabia. This direction is special in Islamic rituals and religious law because it must be faced by Muslims during daily prayers and other religious contexts. The determination of qibla was an important problem for Muslim communities throughout history, because Muslims are required to know the qibla to perform their daily prayers, and because it is needed to determine the orientation of mosques. When Muhammad lived among the Muslims in Medina, he prayed due south, according to the known direction of Mecca. Within the few generations after Muhammad's death in 632, Muslims had reached places far away from Mecca, presenting the problem of determining the qibla in new locations. Initially, Muslims relied on traditional, folk knowledge-based methods, but after the introduction of astronomy in the Islamic world, solutions based on mathematical and astronomical knowledge began to be developed since the early ninth century. The shadow observation method is attested since at least the thirteenth century.Apparent motions of the Sun
Places on earth experience the apparent diurnal motion of the Sun from the east to the west, including a culmination when the Sun reaches its highest point of the day and crosses the local meridian. The Sun also appears to move seasonally between the Tropic of Cancer and the Tropic of Capricorn, therefore the solar culmination in a given place and day usually does not happen directly on the zenith, but to the north or to the south of the zenith. For locations between the two tropics, on certain periods of the year the Sun cross the local latitude, and only then the Sun culminates overhead, and the location is the subsolar point. The Ka'bah is located on 21°25’ north, inside the zone that experiences this phenomenon. In the terminology of Islamic astronomical science, these events are called the "great culmination".Observation
The great culmination in the Ka'bah occurs twice a year, on 27 or 28 May at approximately 12:18 Saudi Arabia Standard Time, and on 15 or 16 July at 12:27 SAST, coinciding with the solar noon and the Zuhr adhan in Mecca. As the sun crosses almost directly above the Ka'bah, of any shadow of the Sun cast by vertical objects on earth will point directly away from the Muslim holy site. The Ka'bah itself casts nearly no shadow. This allows the qibla to be determined by anyone without needing to perform calculations or to use sophisticated instruments. This observation is called rasd al-qibla. As it requires the Sun to be visible, this observation is not possible in the hemisphere opposite the Ka'bah, in which the events occur at local night time. This hemisphere includes most of the Americas, the Pacific Ocean, Australia, and Eastern Indonesia. Instead, these places can observe a comparable event, when the Sun passes directly below the Ka'bah at night. The shadows cast during these times point to the exact opposite of the direction shown during the rasd al-qibla. The antipodal events occur on 12, 13, or 14 January at 00:30 SAST, and again on 28 or 29 November 00:09 SAST. On any of these events, observations made within five minutes' interval, or at the same time one or two days before or after are still accurate with negligible deviation.A practical problem occurs in locations whose angular distances to Mecca are almost equal to 90°. The rasd al-qibla events always occur close to sunrise or sunset. This is the case for several places in the east coast of North America: for instance, the first rasd al-qibla occurs 6 minutes after sunrise in Boston and Montreal, 2 minutes before sunrise in Ottawa, and 11 minutes before in New York City. Therefore, New York City and Ottawa cannot observe it as the Sun is still below the horizon, while in Boston and Montreal, even though the Sun already rises it is still so low that an observation requires a place completely unobstructed by buildings or terrain.
Daily observation
Other than the twice yearly rasd al-qibla, in any location usually there is a moment each day when the Sun points to the qibla, or its exact 180° opposite, because the Sun crosses the direct path between the location and the Ka'bah. However, the time of this daily event varies by place and by the day of the year. The timing can be determined using geographical data and calculations, but for a non-technical person this is much less practical than the yearly rasd al-qibla, the timings of which are the same globally and can be known without making calculations.History
The method of observing the qibla by shadows was attested by the Central Asian astronomer Jaghmini, who wrote that it can be done twice a year when the sun's position in the ecliptic is at 7°21′ Gemini and 22°39′ Cancer. Subsequently, Nasir al-Din al-Tusi also related this method in his work al-Tadhkira al-Nasīriyya fī ʿilm al-Hayʾa, albeit with less precision than Jaghmini:Similar to Jaghmini, Al-Tusi states the two rasd al-qibla days by specifying the sun's position in the ecliptic, instead of any specific date. This is because during their time, the Islamic World used the lunar calendar instead of the solar one, therefore the two days could not be specified in a fixed day and month of the year. Because the obliquity of the ecliptic is slowly decreasing, the values during Jaghmini and al-Tusi's times differ from today's values. As of 2000, the appropriate solar positions are 6°40′ Gemini and 23°20′. Other than specifying the sun's position, the passage by al-Tusi above describes how to convert the noontime in Mecca to the local time.