We know how the Jewish calendar is observed. That is clear... But we do not have a Sanhedrin at this time and we use a calculation which is correct according to the phase of the moon.
According to the Jewish belief the following is true.... Regardless of what karaites do. We know that they are heretics and deny the oral tradition. So I do not accept the explanation you provided.
http://www.chabad.org/library/article_cdo/aid/55194/jewish/Introduction.htmSince Biblical times, various astronomical phenomena have been used to establish uniquely Jewish definitions for the day and its hours, the months and the year.
The length of days and hours vary by the season, controlled by the times of sunset, nightfall, dawn and sunrise. The months and years of the Jewish calendar are established by the cycles of the moon and the sun.
Though the months follow the lunar cycle, the lunar months must always align themselves with the seasons of the year, which are governed by the sun. Thus, the Jewish calendar is "Luni-Solar." The discrepancy between the solar year (365 days) and the lunar year (354 days) was resolved by every so often adding a thirteenth month to the year, to form a "leap year."
In the early times of our history, the High Court (Sanhedrin) in Jerusalem was assigned the tasks of determining the beginning of each month and the balancing of the solar with the lunar years. They relied on direct observation of the New Moon, astronomical data, and other considerations.
In the fourth century after the Temple's destruction, however, when oppression and persecution threatened the continued existence of the Court, a fixed calendar was instituted -- based on the Sanhedrin's closely guarded secrets of calendric calculation. This is the permanent calendar according to which the New Moons and festivals are calculated and celebrated today by Jews all over the world.
Like the original system of observation, it is based on the Luni-Solar principle. It also applies certain rules by which complex astronomical calculations are combined with the religious requirements into an amazingly precise system.
The following pages will provide a brief digest of the factors which control the determination of the Jewish hour, day, month and year.
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http://www.chabad.org/library/article_cdo/aid/526874/jewish/The-Jewish-Month.htmThe Lunar Cycle
The Jewish calendar is based on lunar cycles.1 Towards the beginning of the moon’s cycle, it appears as a thin crescent. That is the signal for a new Jewish month. The moon grows until it is full, the middle of the month, and then it begins to wane until it cannot be seen. It remains invisible for approximately two days2—and then the thin crescent reappears, and the cycle begins again.
The entire cycle takes approximately 29½ days.3 Since a month needs to consist of complete days, a month is sometimes twenty-nine days long (such a month is known as chaser, “missing”), and sometimes thirty (malei, “full”).
Knowing exactly when the month begins has always been important in Jewish practice, because the Torah schedules the Jewish festivals according to the days of the month.
The first day of the month, as well as the thirtieth day of a malei month, is called Rosh Chodesh, the “Head of the Month,” and has semi-festive status. See Why is Rosh Chodesh sometimes one day and sometimes two?
The Jewish Months
Nissan is the first month on the Jewish calendar. Before the Jews left Egypt, on the first day of the month of Nissan, G‑d told Moses and Aaron: “This chodesh (new moon, or month) shall be to you the head of months.”4 Thus the peculiarity of the Jewish calendar: the year begins on Rosh Hashanah, the first day of the month of Tishrei (the anniversary of the creation of Adam and Eve), but Tishrei is not the first month. Rosh Hashanah is actually referred to in the Torah as “the first day of the seventh month.”5
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Sanctifying the Month
“The L‑rd spoke to Moses and to Aaron in the land of Egypt, saying, ‘This chodesh shall be to you the head of months.’” (Exodus 12:1–2)
From the wording of this verse, “shall be to you,” the sages deduced that the responsibility of pinpointing and consecrating the chodesh, the crescent new moon, was entrusted to the leaders of our nation, the Sanhedrin, the rabbinical supreme court of every generation.
Originally, there was no fixed calendar. There was no way to determine in advance the exact day of a coming holiday or bar mitzvah, because there was no way to determine in advance when the month would begin. Each month anew, the Sanhedrin would determine whether the month would be 29 or 30 days long—depending on when the following month’s new moon was first sighted—and would sanctify the new month.
Nowadays
In the 4th century CE, the sage Hillel II foresaw the disbandment of the Sanhedrin, and understood that we would no longer be able to follow a Sanhedrin-based calendar. So Hillel and his rabbinical court established the perpetual calendar which is followed today.
According to this calendar, every month of the year, except for three, has a set number of days:
Nissan—30
Iyar—29
Sivan—30
Tamuz—29
Menachem Av—30
Elul—29
Tishrei—30
Mar Cheshvan—29 or 30
Kislev—29 or 30
Tevet—29
Shevat—30
Adar—29 (in leap years, Adar I has 30 days)
Regarding the variable months of Kislev and Cheshvan, there are three options: 1) Both can be 29 days (the year is chaser), 2) both are 30 (the year is malei), or 3) Cheshvan is 29 and Kislev is 30 (the year is k’sidran, meaning these two months follow the alternating pattern of the rest of the months). Hillel also established the rules that are used to determine whether a year is chaser, malei, or k’sidran.
The rules of the perpetual calendar also ensure that the first day of Rosh Hashanah will never take place on Sunday, Wednesday or Friday.6
When Hillel established the perpetual calendar, he sanctified every Rosh Chodesh until Moshiach will come and reestablish the Sanhedrin.
The Sanhedrin Sanctification
The following is a brief description of the procedure the Sanhedrin followed in days of yore to determine the date of the onset of a new month.
On the 30th day of every month,7 the Sanhedrin would “open for business” in a large courtyard in Jerusalem called Beit Ya’azek. Witnesses who claimed to have seen the new moon on the previous night would come to give their testimony and be cross-examined.8
The members of the Sanhedrin were well schooled in astronomy. They knew exactly when the new moon would have appeared, and where it would have been visible. Nevertheless, the sanctification of the moon depends on the crescent new moon actually being seen by two witnesses. The word “this” (in the above-quoted verse, “This month shall be to you . . .”) implies something that is actually seen.
The rabbis of the Sanhedrin would question the witnesses in the order of their arrival. They knew what the proper responses to their questions ought to be, and were thus quickly able to identify fraudulent claims. Starting with the elder of each pair, they would ask:9 “Tell us how you saw the moon:
In which direction was it in relation to the sun?10
Was it to the north or south?
How high in the sky did the moon appear to be?
In which direction were the crescent’s tips facing?
How wide was it?”
After they had finished questioning the first witness, they would bring in his partner and question him in similar fashion. If the two accounts corroborated, the evidence was accepted.11
That day, the thirtieth day, was now declared Rosh Chodesh of the new month. The head of the Sanhedrin would proclaim: “Mekudash!” (“Sanctified!”) and everyone would respond, “Mekudash! Mekudash!” The previous month was now retroactively determined to have had only twenty-nine days.
Publicizing the New Month
The following night (the second night of the month), huge bonfires were lit on designated mountaintops. Lookouts stationed on other mountaintops would see that a fire had been lit, and would light their own fires. This chain of communication led all the way to Babylonia, so that even very distant communities knew that the day beforehand had been declared Rosh Chodesh.
Eventually, the Sadducees12 started lighting fires on the wrong days in order to manipulate the calendar. To prevent this confusion, the fire-on-mountaintop method of communication was discontinued, and instead messengers were dispatched to Babylonia and all other far-flung Jewish settlements. This took a lot longer, a delay which had (and still has) halachic implications with regards to observance of the second day of holidays in the Diaspora. (See Why are holidays celebrated an extra day in the Diaspora?)
The 30-Day Month
If no witnesses came on the thirtieth day—either because the moon had not been “reborn” yet, or because it was not visible—then the next day, the thirty-first day, was automatically declared Rosh Chodesh, retroactively rendering the previous month a malei month.13
Members of the Sanhedrin would go to a highly visible location, where they would partake in a celebratory meal to signify the new month. No fires were lit that night. The new month is always either on the 30th or 31st day; if they hadn’t lit fires the night before, it was understood that the new month started on the 31st day.
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