For example, the astronomical molad for Cheshvan is 8:44 PM October 7. The moon times for October 8 are 6:12 AM rising and 5:26 PM setting. The noon % illumination is .8% at 11:52. The sunrise and sunset times are 5:37 and 5:16. Thus, it would probably not be visible at moonrise. The noon illumination on October 9 is 4.1 % at 12:49. I would therefore guess that the visibility during this time would be about 1%. This may be visible during the ten minutes after sunset and before moonset (about 17 hours after the astronomical molad). This would mean that Beis Din could not declare the chodesh until October 9 (since they must do so during the day). This would mean that the declared month would match the calculated month and Tishrei would have 30 days as the printed calendars show.
Kislev has the astronomical new moon at 6:52 AM on November 6, which seems to imply that eidus between moonset and sunset on November 6 might be possible however moonset and sunset are both 4:45 PM so it would probably be drowned out. If it was visible, Cheshavan would be a 29 day month, while if not (as is most probable) it would be a 30 day month as shown in the printed calendar
This seems to show that for at least Cheshvan and Kislev, the Sanhedrin would most probably declare the chodesh to be the same as our calculated calendar. It might be interesting to estimate what might happen during the rest of the months of the year and see what would have to be done to ensure that next Rosh Hashannah did not come out on one of the "forbidden" dates.
There are sites that calculate the percent visibility of the moon for a specific date and time. For example, the U.S. Naval observatory says
Although the date and time of each New Moon can be computed exactly (see, for example, Phases of the Moon in Data Services), the visibility of the lunar crescent as a function of the Moon's "age" - the time counted from New Moon - depends upon many factors and cannot be predicted with certainty. In the first two days after New Moon, the young crescent Moon appears very low in the western sky after sunset, and must be viewed through bright twilight. It sets shortly after sunset. The sighting of the lunar crescent within one day of New Moon is usually difficult. The crescent at this time is quite thin, has a low surface brightness, and can easily be lost in the twilight. Generally, the lunar crescent will become visible to suitably-located, experienced observers with good sky conditions about one day after New Moon. However, the time that the crescent actually becomes visible varies quite a bit from one month to another. The record for an early sighting of a lunar crescent, with a telescope, is 12.1 hours after New Moon; for naked-eye sightings, the record is 15.5 hours from New Moon. These are exceptional observations and crescent sightings this early in the lunar month should not be expected as the norm.
Obviously, the visibility of the young lunar crescent depends on sky conditions and the location, experience, and preparation of the observer. Generally, low latitude and high altitude observers who know exactly where and when to look will be favored. For observers at mid-northern latitudes, months near the spring equinox are also favored, because the ecliptic makes a relatively steep angle to the western horizon at sunset during these months (tending to make the Moon's altitude greater).
I would guess that they must have had some occurrance that managed to make it within 15.5 hours (in perfect conditions) in order to set the record. I saw that they did not say when the record was set. Phil Chernofsky has told me that the Israel New Moon Society uses a figure of 18 - 20 hours after the conjuction (astronomical molad) for the earliest possible visibility. I think that if we use the 18 hour figure to get the % of illumination we could calculate. Given that 29.5 days is 708 hours, 18 - 20 hours is approximately 5% - 6% illuminated. We can then use the illumination table to estimate if the moon would be seen around moonset on a particular day. This assumes that the eidim would see the moon between sunset and moonset only. Thus, I would add 18 hours to the conjunction and go to the next sunset time for that estimate.
The Naval observatory calculator generates a table for the year showing the percent illuminated for each day at midnight or noon in Universal Time or one of the U.S. timezones. This would be 7 hours earlier than the time shown for Yerushalayim. The South African Astronomical Observatory has a table of predictions showing where the moon would be visible as well as when it is first visible in South Africa. Another calculator is found at the fishing site http://www.noreast.com/moon/ This calculates the % illuminates at a set hour for a given day. It appears that this is also Eastern Time. Because the Muslims still use "eidus" as the start of their months, they have developed algorithms and maps showing where the moon would be visible.
A set of predictions and maps can be seen at HM Nautical Almanac Office This shows the best time to see the previous new moon for the location requested at the standard time for the first three days of the new moon. It also gives an estimate of the ease of the sighting. For example, September 8 shows the best tim at 5:48 PM with a classification of F. September 9 is 6:04 PM with a classification of C, September 10 is 6:20 PM with a classification of A. The following month is shown with a set of three maps showing the area of the world which can see the moon, but it is more difficult to read. There is no data other than those two months.
There is a Java Applet which gives the location, moon age, and illumination for specific times at specific locations. Yerushalayim is not listed but the latitude (32.083333 N) and longitude (34.8 E) can be entered explicitly. Use the time about halfway (4/9) between sunset and moonset for the estimate being used.
Month Best time Pct Age
Cheshvan Oct 8, 1721 1.7% 1d 02h 33m
Kislev Nov 7, 1651 3.5% 1d 15h 55m