The opening line of this tablet reads: “Year 37 of Nebukadnezar, king of Babylon.”16 Thereafter, it contains detailed descriptions of the position of the moon and planets in relation to different stars and constellations. Also included is one lunar eclipse. Scholars say that all these positions occurred in 568/567 B.C.E., which would make the 18th year of Nebuchadnezzar II, when he destroyed Jerusalem, 587 B.C.E. But do these astronomical references irrefutably point only to the year 568/567 B.C.E.? The tablet mentions a lunar eclipse that was calculated as occurring on the 15th day of the third Babylonian month, Simanu. It is a fact that a lunar eclipse occurred on July 4 (Julian calendar) of this month during 568 B.C.E. However, there was also an eclipse 20 years earlier, on July 15, 588 B.C.E.17 If 588 B.C.E. marked the 37th year of Nebuchadnezzar II, then his 18th year would be 607 B.C.E.—the very year indicated by the Bible’s chronology for the destruction of Jerusalem! (See the time line below.) But does VAT 4956 provide further corroborating evidence for the year 607 B.C.E.? In addition to the aforementioned eclipse, there are 13 sets of lunar observations on the tablet and 15 planetary observations. These describe the position of the moon or planets in relation to certain stars or constellations.18 There are also eight time intervals between the risings and settings of the sun and the moon.18a Because of the superior reliability of the lunar positions, researchers have carefully analyzed these 13 sets of lunar positions on VAT 4956. They analyzed the data with the aid of a computer program capable of showing the location of celestial bodies on a certain date in the past.19 What did their analysis reveal? While not all of these sets of lunar positions match the year 568/567 B.C.E., all 13 sets match calculated positions for 20 years earlier, for the year 588/587 B.C.E. One of the places where the lunar observations fit 588 B.C.E. even better than 568 B.C.E. is shown in the tablet reproduced on these pages. On line 3 of that tablet, we read that the moon was in a certain position on the “night of the 9th [of Nisanu].” However, the scholars who first dated the event to 568 B.C.E. (astronomical -567) acknowledged that in 568 B.C.E., the moon was in that position on “the 8th of Nisanu and not on the 9th.” To support dating the tablet to 568 B.C.E., they postulated that the scribe erroneously wrote “9” instead of “8.”20 But the lunar position in line 3 finds an exact match on Nisanu 9 of 588 B.C.E.21 Clearly, much of the astronomical data in VAT 4956 fits the year 588 B.C.E. as the 37th year of Nebuchadnezzar II. This, therefore, supports the date of 607 B.C.E. for Jerusalem’s destruction—just as the Bible indicates.  jw.org

The opening line of this tablet reads: “Year 37 of Nebukadnezar, king of Babylon.” Thereafter, it contains detailed descriptions of the position of the moon and planets in relation to different stars and constellations. Also included is one lunar eclipse. Scholars say that all these positions occurred in 568/567 B.C.E., which would make the 18th year of Nebuchadnezzar II, when he destroyed Jerusalem, 587 B.C.E. But do these astronomical references irrefutably point only to the year 568/567 B.C.E.? 

The tablet mentions a lunar eclipse that was calculated as occurring on the 15th day of the third Babylonian month, Simanu. It is a fact that a lunar eclipse occurred on July 4 (Julian calendar) of this month during 568 B.C.E. However, there was also an eclipse 20 years earlier, on July 15, 588 B.C.E. 

If 588 B.C.E. marked the 37th year of Nebuchadnezzar II, then his 18th year would be 607 B.C.E.—the very year indicated by the Bible’s chronology for the destruction of Jerusalem! (See the time line below.) But does VAT 4956 provide further corroborating evidence for the year 607 B.C.E.? 

In addition to the aforementioned eclipse, there are 13 sets of lunar observations on the tablet and 15 planetary observations. These describe the position of the moon or planets in relation to certain stars or constellations.18 There are also eight time intervals between the risings and settings of the sun and the moon. 

Because of the superior reliability of the lunar positions, researchers have carefully analyzed these 13 sets of lunar positions on VAT 4956. They analyzed the data with the aid of a computer program capable of showing the location of celestial bodies on a certain date in the past. What did their analysis reveal? While not all of these sets of lunar positions match the year 568/567 B.C.E., all 13 sets match calculated positions for 20 years earlier, for the year 588/587 B.C.E. 

One of the places where the lunar observations fit 588 B.C.E. even better than 568 B.C.E. is shown in the tablet reproduced on these pages. On line 3 of that tablet, we read that the moon was in a certain position on the “night of the 9th [of Nisanu].” However, the scholars who first dated the event to 568 B.C.E. (astronomical -567) acknowledged that in 568 B.C.E., the moon was in that position on “the 8th of Nisanu and not on the 9th.” To support dating the tablet to 568 B.C.E., they postulated that the scribe erroneously wrote “9” instead of “8.” But the lunar position in line 3 finds an exact match on Nisanu 9 of 588 B.C.E. 

Clearly, much of the astronomical data in VAT 4956 fits the year 588 B.C.E. as the 37th year of Nebuchadnezzar II. This, therefore, supports the date of 607 B.C.E. for Jerusalem’s destruction—just as the Bible indicates. 
When Was Ancient Jerusalem Destroyed?​—Part Two – jw.org

The return date is pretty undisputed- 537 BCE. And the bible specifies an exile of 70 years. So, a 70 year exile ending in 537 BCE would have to start in 607 BCE. Secular sources don't mind at all refuting the bible, so they're OK with a 50 year exile starting in 587... but as many people cite this date as more authoritative than scripture... most people just know "that's the date we're told (sure that someone has some sort of physical evidence)."

The physical evidence is a babylonian tablet labeled VAT 4956

The opening line of this tablet reads: “Year 37 of Nebukadnezar, king of Babylon.”16 Thereafter, it contains detailed descriptions of the position of the moon and planets in relation to different stars and constellations. Also included is one lunar eclipse. Scholars say that all these positions occurred in 568/567 B.C.E., which would make the 18th year of Nebuchadnezzar II, when he destroyed Jerusalem, 587 B.C.E. But do these astronomical references irrefutably point only to the year 568/567 B.C.E.? The tablet mentions a lunar eclipse that was calculated as occurring on the 15th day of the third Babylonian month, Simanu. It is a fact that a lunar eclipse occurred on July 4 (Julian calendar) of this month during 568 B.C.E. However, there was also an eclipse 20 years earlier, on July 15, 588 B.C.E.17 If 588 B.C.E. marked the 37th year of Nebuchadnezzar II, then his 18th year would be 607 B.C.E.—the very year indicated by the Bible’s chronology for the destruction of Jerusalem! (See the time line below.) But does VAT 4956 provide further corroborating evidence for the year 607 B.C.E.? In addition to the aforementioned eclipse, there are 13 sets of lunar observations on the tablet and 15 planetary observations. These describe the position of the moon or planets in relation to certain stars or constellations.18 There are also eight time intervals between the risings and settings of the sun and the moon.18a Because of the superior reliability of the lunar positions, researchers have carefully analyzed these 13 sets of lunar positions on VAT 4956. They analyzed the data with the aid of a computer program capable of showing the location of celestial bodies on a certain date in the past.19 What did their analysis reveal? While not all of these sets of lunar positions match the year 568/567 B.C.E., all 13 sets match calculated positions for 20 years earlier, for the year 588/587 B.C.E. One of the places where the lunar observations fit 588 B.C.E. even better than 568 B.C.E. is shown in the tablet reproduced on these pages. On line 3 of that tablet, we read that the moon was in a certain position on the “night of the 9th [of Nisanu].” However, the scholars who first dated the event to 568 B.C.E. (astronomical -567) acknowledged that in 568 B.C.E., the moon was in that position on “the 8th of Nisanu and not on the 9th.” To support dating the tablet to 568 B.C.E., they postulated that the scribe erroneously wrote “9” instead of “8.”20 But the lunar position in line 3 finds an exact match on Nisanu 9 of 588 B.C.E.21 Clearly, much of the astronomical data in VAT 4956 fits the year 588 B.C.E. as the 37th year of Nebuchadnezzar II. This, therefore, supports the date of 607 B.C.E. for Jerusalem’s destruction—just as the Bible indicates. (https://www.jw.org/en/library/magazines/wp20111101/When-Was-Ancient-Jerusalem-Destroyed-Part-Two/)

The strangest outlier come from Jewish sources that put the babylonian siege starting at 425 BCE, with the temple destroyed in 423 BCE, and the return to israel at 353 BCE.

https://www.chabad.org/library/article_cdo/aid/3915966/jewish/Timeline-of-Jewish-History.htm

The return date is pretty undisputed- 537 BCE. And the Bible specifies an exile of 70 years. So, a 70 year exile ending in 537 BCE would have to start in 607 BCE. Secular sources don't mind at all refuting the Bible, so they're OK with a 50 year exile starting in 587... but as many people cite this date as more authoritative than scripture... most people just know "that's the date we're told (sure that someone has some sort of physical evidence)."

The physical evidence is a Babylonian tablet labeled VAT 4956

The opening line of this tablet reads: “Year 37 of Nebukadnezar, king of Babylon.”16 Thereafter, it contains detailed descriptions of the position of the moon and planets in relation to different stars and constellations. Also included is one lunar eclipse. Scholars say that all these positions occurred in 568/567 B.C.E., which would make the 18th year of Nebuchadnezzar II, when he destroyed Jerusalem, 587 B.C.E. But do these astronomical references irrefutably point only to the year 568/567 B.C.E.? The tablet mentions a lunar eclipse that was calculated as occurring on the 15th day of the third Babylonian month, Simanu. It is a fact that a lunar eclipse occurred on July 4 (Julian calendar) of this month during 568 B.C.E. However, there was also an eclipse 20 years earlier, on July 15, 588 B.C.E.17 If 588 B.C.E. marked the 37th year of Nebuchadnezzar II, then his 18th year would be 607 B.C.E.—the very year indicated by the Bible’s chronology for the destruction of Jerusalem! (See the time line below.) But does VAT 4956 provide further corroborating evidence for the year 607 B.C.E.? In addition to the aforementioned eclipse, there are 13 sets of lunar observations on the tablet and 15 planetary observations. These describe the position of the moon or planets in relation to certain stars or constellations.18 There are also eight time intervals between the risings and settings of the sun and the moon.18a Because of the superior reliability of the lunar positions, researchers have carefully analyzed these 13 sets of lunar positions on VAT 4956. They analyzed the data with the aid of a computer program capable of showing the location of celestial bodies on a certain date in the past.19 What did their analysis reveal? While not all of these sets of lunar positions match the year 568/567 B.C.E., all 13 sets match calculated positions for 20 years earlier, for the year 588/587 B.C.E. One of the places where the lunar observations fit 588 B.C.E. even better than 568 B.C.E. is shown in the tablet reproduced on these pages. On line 3 of that tablet, we read that the moon was in a certain position on the “night of the 9th [of Nisanu].” However, the scholars who first dated the event to 568 B.C.E. (astronomical -567) acknowledged that in 568 B.C.E., the moon was in that position on “the 8th of Nisanu and not on the 9th.” To support dating the tablet to 568 B.C.E., they postulated that the scribe erroneously wrote “9” instead of “8.”20 But the lunar position in line 3 finds an exact match on Nisanu 9 of 588 B.C.E.21 Clearly, much of the astronomical data in VAT 4956 fits the year 588 B.C.E. as the 37th year of Nebuchadnezzar II. This, therefore, supports the date of 607 B.C.E. for Jerusalem’s destruction—just as the Bible indicates. jw.org

The strangest outlier come from Jewish sources that put the Babylonian siege starting at 425 BCE, with the temple destroyed in 423 BCE, and the return to israel at 353 BCE. Chabad.org