The Chai Tables

(last revised 06/21/2009)


I. Introduction

Tables for sunrise are of course not a new thing. The unique feature of the Chai Tables is they are the first tables for North America and Europe to include visible sunrise times (and visible sunset times of the USA). What does this mean, and why is this significant? There are at least three ways to define the sunrise. The first one is called the astronomical sunrise. It corresponds to the time the sun appears to an observer at his local height if all the mountainous obstructions in the horizon are disregarded (blue arrow). The second type of sunrise is called the mishor sunrise. It corresponds to the time that the sun is seen rising over a flat horizon for an observer at the the height of the horizon (green arrow). This is the sunrise that is published by the Naval Observatories and is printed by the newspapers in the USA. The third type of sunrise is the visible sunrise, corresponding to the time that the sun is actually seen to rise over the true horizon (black arrow).   The definitions for sunset are the same (except the figure is reflected about the observer).

Which is the proper sunrise to use for the purpose of morning prayer? The gemorah Yerushalmi in tractate Brochos (first chapter) states in the name of Rebbi Zavdi in the name of Rebbi Yonah that sunrise (for purposes of krias shma) is when the sun's rays begin to illuminate the tops of the mountains. In the same place, the gemorah relates how Rebbi Elazar ben Azaria and Rebbi Akiva waited to say krias shma until the sun appeared on the top of the mountains. The Bi'ur Halachah (Orach Chaim, chapter 58) explains that the gemorah is rendering the halachic opinion that the time of sunrise for prayer purposes is when the sun becomes visible on the horizon. The gemorah Bavli in tractate Brochos (p. 7) describes how Hashem is angry during one special moment of the day, i.e., at sunrise, when all the kings of the nations bow down to the rising sun. Of course, they bowed down to a visible sun.

The rabbis, of blessed memory, tell us in tractate Megillah (page 20) that the miztvos that are done during the day, are permitted to be done, a priori, only after the sunrise even though the day begins at dawn. Rashi explains that this is so we should have no doubt that the night has passed. The Ran explains further that not everyone knows how to distinguish the dawn from the night. We therefore wait until sunrise for only then is it clear to everyone that the night has passed. The Ran seems to imply that only when the sun is visible (i.e., after the visible sunrise) can there be no doubt that night has passed. The Mishnah Brurah (in Section 588(2) ) seems to understand that this was the intention of the Ran . The Mishnah Brurah, when discussing the proper time to blow shofar, first brings down the Ran (above) and then quotes the Pri Megodim that one doesn't have to wait to blow shofar until the entire sun is visible, rather it is sufficient to wait until only the very first part of the sun is visible. Once again, the Mishnah Brurah seems to have decided that the proper sunrise for mitzvos hayom purposes is the visible sunrise. This is also in accordance with the mishnoh in tractate Yumah (p. 37) which describes the nivreshes that Queen Heleni contributed to the Beis Hamikdosh. The nivreshes reflected the sun's rays at the moment of the visible sunrise in order that everybody would know that the time of krias shma had arrived. (Tosofos (ibid.) explains that the vasikim were able to determine this time without the help of the nivreshes, and therefore finished krias shma right before the visible sunrise, at which time they began the shmonah esray.)

However, the gemorah Bavli tractate Pesachim (p. 93) states that the interval between sunset and twilight is the same as the interval between dawn and sunrise. This is clearly the astronomical sunrise (since the interval between the visible sunrise and dawn is not necessary equal to the interval between the sunset and twilight). In fact, this is not a contradiction to the gemorahs we have brought above. The gemorah in Pesachim is concerned with how one calculate times of the day (shaos zemanios) for which one uses the astronomical or mishor sunrise. However the gemorahs we brought above were concerned with the proper times for krias shma and shmonah esray. For these purpose one must wait until the visible sunrise. Why is that so? The Meiri on tractacte Brochos (p. 9 second side) explains that the awareness of Hashem generated by the shmonah esreh balances the lack of awareness of Hashem as displayed by most nations. The Jewish people therefore accustomed themselves to pray at the same time as those nations begin their sun worship, i.e., at the moment that the sun became visible.

It is probably for the reasons given above that the custom of Eretz Yisroel is to wait to start the shmonah esreh until the moment of the visible sunrise, v'mezion tezei torah. The purpose of these tables is to give this option to other Jewish communities.

In practice how does one define the visible sunrise for cities built in valleys and on tops of hills, as is the case of most cities in the world. Usually the high places will see the sunrise before the places built in valleys. Do all the people in the city use the sunrise seen on the high places or does each one have his own visible sunrise? As mentioned above, the mishnoh in Yumah (p. 37) describes how Queen Heleni contributed a golden menorah ("nivreshes") to the second Temple in order to reflect the rays of the sun during the initial minutes of sunrise (see the top figure on the right). The gemorah (ibid) brings a breisoh stating that in this way everybody in Jerusalem would know that the time for krias shma had arrived. It is known that even in the second Temple era, most of Jerusalem extended in the direction of the Kidron and Himnom valleys. The city then extended from these places of low elevation northward and westward to higher terrain. Not everyone in Jerusalem would have seen the sunrise at this time. The visible sunrise seen by the inhabitants of the lower portions of Jerusalem for the summer months was at 15 minutes later than the visible sunrise seen by the nivreshes if it was placed on the top of the Temple as seems to be the opinion of the Tosofos and the Ritvah (middle and bottom figures on the right). Nevertheless there was only one sunrise time for the entire city, and it was the earliest visible sunrise to be seen in the city. This reflects the halochic principle that we don't make divisions within one city ("ein osim agudos agudos bair achas"). Another example of this is in the gemorah Shabbos 118b which relates how Rebbi Yosi lauded all the inhabitants of Tiberius for accepting Shabbos early. This implies that they began Shabbos together otherwise he would have lauded those that accepted the earliest Shabbos time in the city. This is despite the fact that Tiberius is built on the side of a mountain and the times of sunset varies greatly between parts of the city. (These proofs for using the earliest visible sunrise time for the entire city is brought in part by the Tshuvos "Sifsei Yeshonim", pp. 26-27, of Rav Moshe Nehemiah Cohanav, a contemporary of Rav Yehoshuah Leib Diskin.)

The visible sunrise is sometimes obscured by very local obstructions. In the most absurd example, an individual in a deep ditch won't see the sunrise until midday. It sometimes occurs that the sunrise horizon for an entire city is obscured by local mountainous obstructions. This was the case of Jerusalem in the late 1800's when the entire population dwelled in what we know today as the Old City. Their sunrise horizon was obstructed for half of the year by Har Hazeisim which is a little more than one kilometer from the eastern end of the Old City. The question of what one does in such a situation was brought to Rav Yehoshuah Leib Diskin. He rendered a pesak din (recorded in "HaNivreshes", p. 58-59) that one certainly discounts very local mountains such as Har Hazeisim, but one need not disregard distant mountains such as the Moab mountains (which are between 40-70 km from Jerusalem). However, no distance was given to define the dividing line between close and far mountains. (It is also unclear how Rav Yehoshuah Leib Diskin reconciled his pesak with the sugiah of the nivreshes, which implies that one does not discount Har Hazeisim. It is possible that Rav Yehoshuah Leib Diskin felt that Har Hazeisim could be ignored in calculating the sunrise times for the Old City for the following reason. The sunrise times seen by the nivreshes, if it was placed on top of the Temple, was probably not significantly different from the sunrise times seen from Har Hazeisim. Therefore any delay to the sunrise time caused by Har Hazeisim, as seen by an observer on the ground,was due to that observer's low elevation in comparison to the height of the nevreshes.) In our tables for Eretz Yisroel (Bikurei YosefTables) we therefore included all mountains in the horizon as permitted by the accuracy of the Digital Terrain Model (DTM). We have followed this procedure for the Chai Tables.

The use of one sunrise time for an entire city works well for a city the size of Jerusalem. To this day, the poskim of Jerusalem agree that all of Jerusalem (which extends about 16 km. north-south, and 15 km. east-west) abides by the earliest sunrise time as seen from the highest places in Jerusalem. (The first visible sunrise is generally not seen at the same place every day, since the sunrise occurs over a different part of the mountainous horizon on different days of the year.) However, how is the sunrise defined for cities the size of New York, Chicago, or Los Angeles where continuous habitation can extend for over 100 km or more! In such a case even the astronomical sunrise differs by about 4 minutes from the most eastern to the most western portion of continuous habitation in that area.

We say that that there must be one sunrise time for the inhabitants of a small city since night and day cannot exist simultaneously at any one small place. Furthermore, it is hard to imagine that any slight change of observer height and effect of local obstructions should change the sunrise time of the observer in a significant manner. If that would be the case, then the sun would never rise or set for someone indoors! Obviously, he must open his windows and take a look. Similarly, he should go up to a nearby high vantage point to observe an earlier sunrise. It is necessary to say that there is no such thing as a local visible sunrise for any small place. However, when night and day does in fact exist simultaneously, in the case of a large city, then certainly we are not restricted to one visible sunrise for everybody.

To our knowledge, this problem has not been dealt with by the poskim. The Chai Tables attempt to solve this problem as follows. For large cities, one must calculate visible sunrises for sub-city areas. In general, a person is associated with a sub-city area which is considered his local environment. His synagogue is probably there, as well as the stores that he frequents. This is his local city, neighborhood, or community. The earliest sunrise time to be seen in that area would also be his sunrise time. The gemorah Eruvin 60b infact implies that the area of a city that a person frequents is equivalent to his local environment. That gemorah states in the name of Rebbi Yeshua Ben Levi that if a person's techum of 2000 amos extends to the end of a city, then that city is considered as only 4 amos long, and he can travel another 1996 amos. Why did Chazal use the value of 4 amos? To tell us that a person's city is equivalent to his own domain (which is of course 4 amos) since he makes use of that area as though it was his own 4 amos. Of course, in the case of sunrise, this area must be limited, as explained above. For convenience, the Chai Tables define this region as a circle about the observer. What would be the dimensions of that circle? Since Jerusalem is considered as one city even though it extends about 8 km. in each direction from the city's center, then it is reasonable to use a radius of 8 km. as the default radius of this circle (The Search Radius). The earliest sunrise within this radius (which could be from different places on different days as explained above) will thereby define the sunrise for any person near the center of that circle. This circle (yellow circle in the figure for the example of West Hollywood, California, USA), of course, moves with the observer. But the sunrise times for neighboring circles separated by distances smaller than 8 km will be similar. This is because it is unlikely that any new high vantage points (red spots in the figure) will be included within these circles for small changes in their center position. In this way, night and day will not reign simultaneously in any one place. In fact, for cities smaller than 16 km in diameter, all residents of the city will have the same visible sunrise, as is case for Eretz Yisroel.

In many cases after averaging over the possible vantage points within the radius of influence, the calculated visible sunrise time will become similar to the mishor sunrise. The only real difference will be for truly high vantage points which will actually see an earlier sunrise than the mishor sunrise, and for those places with high mountains in the distance in all directions. For the last case the visible sunrise can be significantly later than the mishor sunrise.

If a city is nestled between hills, the inclusion of vantage points near the hills can often degrade the quality of the table (since these high vantage points are situated close to even higher obstructions). Therefore in the event that a calculated table of the visible times has many times marked in green (indicating near obstructions), one can try to improve the quality of the table by reducing the "Search Radius".

II. The Visible Sunset

In the gemorah Shabbos 118b ( as mentioned above) Rebbi Yosi lauded the inhabitants of Tiberias for ushering in Shabbos when the sun sank over the visible horizon.  Rashi (ibid) implies that this was only a stringency.  Rather, the true sunset time for Tiberias was the time of sunset for Zippori, located on top of the hill.  We understand from Rashi that sunset is the time when the sun sets for the high places within or adjoining the city.  .Rebeinu Nisim (ibid) seems to agree with Rashi and  wrote further that the visible sunset of Tiberius is a "siag" (fence), and those who accept it are praise worthy.   This is in fact brought down as the halocho by the Magen Avrohom and the Beir Hetev in Orech Chaim, section 233.  That is, the time of minhah is "a priori" up to the time of the visible sunset .  For these reasons, Hagaon Rav Moshe Feinstein wrote in Egros Moshe (Orech Chaim Vol. I, responsa 97) that bnei torah should be stringent to accept a "siag" upon themselves by ushering Shabbos before the time of the visible sunset over the mountains. 

Due to the availability in the US of accurate DTM's, it is possible to provide visible sunset times for the cities of the US.  Even though the 30-m SRTM data is not reliable for many urban areas (see below), the 30-m NED data can be used to obtain reasonably reliable heights within the city.  It is therefore possible to provide reliable visible sunset times for much of the US. (The visible sunset times for Eretz Yisroel are calculated using the 25-m DTM of Israel.)  These times should be used only as a siag, and only after consulting with a competent halochic authority..

III. How is the visible sunrise calculated for any observer

For any city, likely vantage points that would contribute to the earliest sunrise were searched for, and the horizon profiles for those places were calculated and recorded. When you, the user of our program, choose a set of coordinates, the subset of vantage points within the chosen circle of influence is determined, and the sunrise times for all these vantage points are calculated. The earliest sunrise time that is seen (i.e., calculated) among these vantage points determines the sunrise time for the observer. (For details on the calculation of these profiles and their sunrise times, see the paper of Keller, Hall that can be downloaded from the "published papers" section of the Chai Table's home page.)

IV. Accuracy of Visible Sunrise Calculations

In the past, the elevation model that is used for most of the Chai Tables is the United States Geological Survey's (USGS) GTOPO30 Digital Elevation Model of the world. The GTOPO30 consists of an evenly spaced grid of interpolated elevation readings approximately every kilometer. The topographic data base that was the source of the GTOPO30 for the USA was the 30 meter topographic maps of the USGS. The source of the GTOPO30 for most of Eurasia, and large parts of Africa, South America, Mexico, Canada, and Central America was the Digital Terrain Elevation Data (DTED) Model produced by the USA's National Imaging and Mapping Agency (NIMA). In these regions it is expected that the GTOPO30 has an absolute vertical resolution of 30 meters, and a root mean square vertical accuracy of 18 meters. The coarseness of this grid means that it is not very useful in determining horizon profiles if there are obstructions within 30 kilometers. This holds true especially for relatively flat areas. Due to this fact, very local obstructions (closer than 5 km to any observation point) were not included in the calculations. However, when many vantage points are averaged over the circle of influence with a radius equal to 8 km., then inaccuracies of the GTOPO30 should not cause more than a minute error in the sunrise times. This minute inaccuracy was determined by comparing their times to visible sunrise times calculated from the USGS 100 meter Digital Elevation Model of the USA.

A preliminary version of the Shuttle Radar Topography Mission (SRTM) digital terrain model encompassing most of the world has been released.  For the USA, the data set consists of equally spaced heights approximately every 30 meters (SRTM-Level 2, i.e., SRTM-2).  The SRTM DTM data spacing for the rest of the world is one elevation value for approximately every hundred meters (SRTM-Level 1, i.e., SRTM-1).  At present, almost all of the Chai tables have been recalculated using the SRTM.

A comparison of the SRTM-1 elevations to known elevations in one location in Canada has shown that the vertical resolution is (+/-) 6 meters, while horizontal (spatial) resolution is excellent (B. Rabus, "First experiences with validating SRTM X-band data over Canada").  This compares favorably with the expected vertical resolution of (+/-)10 meters.  Therefore, the horizon profiles calculated from the SRTM-2 should be accurate to approximately 20 sec. as long as obstructions are further than 6 km. from the observer.  Assuming an average distance to obstructions of 20 km, the accuracy of SRTM-2 data will increase to approximately 5 seconds. SRTM-1 data's horizon profiles will be accurate to 20 seconds for obstructions greater than 18 km.  Therefore, only a 5 seconds have been added to calculated times using SRTM-2 data.  (Twenty  seconds will be added to calculations based on the SRTM-1 terrain model for areas outside the US.)  This cushion in combination with the 15 second cushion added for atmospheric effects (see below) was deemed adequate to account for better than 90% of any expected error.

Due to the greater accuracy of the SRTM DTMs, only 1 km of near obstructions was ignored in the calculations (versus 5 km for the GTOPO30). Times that are as accurate or more accurate than the minimum expected accuracy are printed in black.

Since sunrise times calculated from SRTM data set can be highly accurate, it is appropriate to notify the user when the calculation is in fact degraded due to close obstructions.   Therefore, times that are likely to be inaccurate due to close obstructions (less than 6 km for the SRTM-1/2) are printed in green. 

An unfortunate degradation of SRTM accuracy occurs within urban areas having a high concentration of tall buildings (see the article by Rabus, ibid.)  This effect is  especially noticeable in flat regions of the world.  In such cases, tall buildings dominate the skyline (and therefore, the horizon profiles).  The buildings appear as spikes (glitches) whose shapes are dependent on the buildings and the distance from the observation point. Tall building can erroneously increase the SRTM ground elevations to more than 30 meters.   If many points in the city contribute to the sunrise time, then the glitches will be averaged out.   However, shorter buildings also add an ubiquitous elevation scatter of between 5-10 meters.  I have overcome this problem in the USA by substituting the SRTM heights in most urban areas with the 30 meter NED (National Elevation Data Set) heights.  In those cases, the SRTM data is only used for determining the shape of the distant horizon.  Unfortunately, there is no affordable alternative to the SRTM for outside the USA. 

A further source of error for the Chai Tables, notwithstanding the DTM used for the calculations,  results from inaccuracies in the modeling of the atmosphere. The atmospheric model I use is based on modified mid-latitude "summer" and "winter" atmospheric models used by the Lowtran 6 Computer Codes. For tropical latitudes, I use the "summer" atmosphere for the entire year. For latitudes above and below N 20 and S 20 degrees, respectively, I use a combination of "winter" and "summer" atmospheres. The percentage of winter days of the year are increased as a function of latitude in a stepwise fashion. For example, in the Northern Hemisphere, all latitudes between N 30 degrees and N 40 degrees are assumed to have a similar summer to winter atmosphere ratio (1.28) as that of Eretz Yisroel (which lies between latitudes of N 30 to N 33 degrees). This ratio was determined from daily observations of the visible sunrise at several places in Eretz Yisroel.

Comparison of actual sunrise observations done in Jerusalem to calculated sunrise times have shown that sunrise times can be calculated for distant horizons (farther than 20 km.) to an accuracy of at least minutes for 90% of the time. (See the figure on the right which is a plot of the visible sunrise minus the astronomical sunrise for the Jerusalem neighborhood Armon Hanatziv. Black dots are calculations, blue dots represent 10 years of observations by Rav Avraham Druk. For more information see the paper by Keller and Hall) Since those calculations used the 25 meter Digital Terrain Model (DTM) of Eretz Yisorel, almost all the discrepancy between calculation and observation was due to atmospheric effects. Therefore, in the case of distant horizons, and for latitudes close to those of Eretz Yisorel, the Chai Tables of visible sunrise times should have a maximum accuracy of minutes. Typically, the accuracy should be on the order of the sum of the atmospheric inaccuracy ( minutes) and the inaccuracy inherent in the SRTM (5 sec. for SRTM-2,  20 seconds for SRTM-1)  and in the GTOPO30 ( minutes). I have therefore added a cushion of minutes to all the calculated visible sunrise times calculated using the GTOPO30, a 20 second cushion to visible sunrise times calculated in the USA using the SRTM-2, and a 35 sec. cushion to visible sunrise times calculated using the SRTM-1 (for outside the USA).  The resultant times are rounded to the nearest later 5 seconds.  (You can contact us to verify that your sunrise horizon is determined by far obstructions.)

The almost exclusive source of maps for my latest recalculation of the horizon profiles is www.expedia.com.  Expedia provide both topographical and road maps.  The former allows for relatively accurate (to better than 1 km) placement of the city center. The latter allows for placement of neighborhoods to better than 1 km.

The maps from Expedia seem to be derived from NIMA's GEOnet database coordinates. Please refer to Expedia's maps for positions of the city centers and neighborhoods. If the user can determine his center coordinates (e.g., by using GPS) then they can be used instead of ours (see below).

V. How do you pick your place?

In order to maximize the user-friendliness of this program, I have provided the names and coordinates of cities that have a significant Jewish Orthodox presence and neighborhoods within these cities. When the user picks one of these neighborhoods or sub-cities, then the program uses the known coordinates of that neighborhood or sub city as the center of the observer's circle of influence. See below for the example of Westwood, Los Angeles area, California, USA. If your metropolitan area is not listed, then you will be only able to calculate mishor and astronomical times.

Since it is difficult to know the names of all the neighborhoods where Orthodox Jews live, I usually list all the sub-cities that are marked by the maps mentioned in section III. If the Jewish neighborhoods are known, then they have been listed even if they do not appear on the maps. (If we have missed your neighborhood, let us know. Please help us make the Chai Tables more useful). In the case that your metropolitan area is listed but your neighborhood has been inadvertently skipped, or if you wish to pick an exact place, then the "Chai Tables Geofinder" button should be used.  This button sends the user to a Google map interface.  With this interface, an exact place can be found.  These coordinates will be used in calculating the center of the search radius.   If desired, the user can input these coordinates by hand (although this is not recommended since the sign convention for longitudes can be confusing).  To use the "Geofinder" (or to input the coordinates by hand) don't pick anything in "Step 2" (i.e., leave it displaying "Select a neighborhood"). Then press the "Geofinder" button (or input your coordinates) in Step 3 (see the figure below).  The program uses the coordinates chosen with help of the "Geofinder" (or inputed by hand) as the center of the circle of influence and searches for vantage points within the radius of the circle of influence. If no vantage points can be found, the user can still calculate the astronomical and/or mishor sunrise and sunset times for his place (the astronomical and mishor sunset are defined like the astronomical and mishor sunrise as described above).

(P.S., If we have missed your Jewish Orthodox community please contact us by filling out the feedback form).


It is the hope of the author that these tables will help spread the proper observance of the mitzvah of tefilas vasikin in the world.


VI. Acknowledgments

The calculations of these tables would not be possible without the generous help of Professor John Hall of the Geological Survey of Israel who provided the author with the computer hardware necessary for these calculations .



A Short Introduction to the Subject of "Zemanei Hayom"

The halochic day is divided into twelve equal hours called "shaos zemanios". These hours correspond to hours of a sun clock and therefore change their length with respect to a mechanical clock during the course of the year. These hours are used in calculating the permissable times for many mitzvos. For example, "krias shma" must be said before the end of the third hour, and morning prayer must be completed before the end of the fourth hour. Even though that the halochic day begins at dawn and ends at twilight, there are several opinions concerning from when and to when the "shaos zemanios" are counted. There are three major opinions.

The Opinion of the "Magen Avrohom" (which is the opinion of the "Trumas Hadeshen")

According to this opinion the "shaos zemanios" are calculated by dividing the time interval between dawn and twilight into 12 equal parts. There are also many opinions on how to calculate dawn and twilight. However, the amount of time between dawn and sunrise and between sunset and twilight are equal (i.e., the time that it takes to walk approximately four kilometers). Usually dawn/twilight is calculated to be approximately 72 or 90 minutes before/after the "mishor" sunrise/sunset. (Note that the 72 or 90 minutes are usually calculated in degrees of solar depresssion. This results in 72 or 90 minutes for Eertz Yisroel during Nisan and Tishrey.) Many people accept upon themselves the stringencies of this opinion in relation to it's time for "sof zman krias shma".

The Opinion of the "Groh" (The Gaon of Vilna) (which is the opinion of the "Levush")

According to this opinion the "shaos zemanios" are calculated by dividing the time interval between sunrise and sunset into 12 equal parts. The sunrise and sunset are taken to be the "astronomical" or "mishor" sunirse and sunset (see above for the definition of "astronomical" and "mishor" sunrise and sunset). Most ashkenazi poskim consider this opinion to be the halochoh.

The Opinion of the "Ben Ish Chai" (which is typically used by Sephardic Jews)

According to this opinion the "shaos zemanios" are calculated by dividing the time interval between dawn and twilight into 12 equal parts. However the time intervals between dawn and sunrise and between sunset and twilight are not equal. This results in an unsymmetrical day. The time interval between dawn and sunrise is larger than the time interval between sunset and twilight. This is due to the fact that the halochic twilight is the time after sunset for three medium stars to appear. This is approximately 20-25 minutes in Eretz Yisroel. However the civil dawn is approximately 90 to 72 minutes before the "mishor" sunrise.

One must consult a competent halochic authority concerning which opinion to practice.