There are three plausible theories about the nature of the astronomical phenomenon that was interpreted by the Magi from the East (astronomers/astrologers from Babylon or Persia) as signalling that a King of the Jews of divine nature had been born.
Theory 1
Triple conjunction of Jupiter and Saturn in 7 BC (occurs every 800 years), in the constellation of Pisces. Proposed by Ferrari-D'Occhieppo (1989) 1 and Parpola (2001) 2. (Dates are from 2.)
1st conjunction - 27 May: rising "in the East", Jupiter first, Saturn soon after.
16 July: Jupiter reached its first stationary point.
2nd conjunction - 6 October: the two planets, 1º apart in longitude, emerged "in the East" at sunset, in opposition to the sun and shining at their brightest, with Jupiter appearing twice as bright as Sirius, the brightest star, and appearing directly above Saturn.
=> The Magi set out on their trip to Palestine (probably from Babylon).
7 November: Jupiter reached its second stationary point.
=> The Magi leave Jerusalem towards Bethlehem in the late afternoon of 7 November 7 BC. Ferrari-D'Occhieppo has a compelling description of how the Magi would have seen Jupiter and the zodiacal light in front of them "standing over" Bethlehem.
(Jesus' family flee to Egypt shortly afterwards and stay there till after Herod's death in late March / early April 4 BC, i.e. for 2.5 years.)
3rd conjunction - 1 December.
Possible objection to this theory: if the Magi told Herod in November that they had first seen the star (Jupiter) "at its rising" in May, a 1-year range would have provided more than ample margin for the age of children to be executed in the Bethlehem area. However, for someone that was willing to execute infants in the first place, doubling the age range from 1 to 2 years and the probable ensuing body count from 10 to 20 children may not have seemed a big deal.
Theory 2
Two occultations ("eclipses") of Jupiter by the Moon in 6 BC in the constellation of Aries (the sign of the Jews). Proposed by Molnar (1999)3.
1st occultation: 20 March. (It was preceded by a lunar occultation of Saturn on 19 March in the constellation of Pisces.4)
2nd occultation: 17 April, when Jupiter emerged "in the East" as a morning star in Aries, with the Sun also in Aries, so the Magi's star was the planet Jupiter in its “heliacal rising”, in the morning, in the East, a fairly precise distance ahead of the sun at dawn, in Aries. (It was preceded by a lunar occulation of Saturn on 16 April in the constellation of Pisces, well within 1º of the first point of Aries.4)
=> The Magi set out on their trip to Palestine (probably from Babylon).
In August Jupiter became stationary and then "went before" (retrograde motion) until it became stationary again ("stood over") on 19 December, back into the sign of Aries.
=> The Magi leave Jerusalem towards Bethlehem in the late afternoon of 19 December 6 BC.
(Jesus' family flee to Egypt shortly afterwards and stay there till after Herod's death in late March / early April 4 BC, i.e. for 1.5 years.)
Possible objection to this theory: if the Magi told Herod in December that they had seen the star (Jupiter) "at its rising" in April, a 1-year range would have provided more than ample margin for the age of children to be executed in the Bethlehem area. However, for someone that was willing to execute infants in the first place, doubling the age range from 1 to 2 years and the probable ensuing body count from 10 to 20 children may not have seemed a big deal.
Theory 3
Combines theory 1 + theory 2.
The Magi took notice of both the events in 7 BC described by theory 1 and those in 6 BC described by theory 2, travelling to Palestine and then to Bethlehem in 6 BC as described in theory 2.
Humphreys (1995) 5 proposes a slightly different version of this theory: the triple conjunction of 7 BC in the constellation of Pisces (plus a subsequent joining by Mars so that in February 6 BC the three planets, still in Pisces, were separated by only about eight degrees) alerted the Magi, and the comet that, as recorded by Chinese sources, was observed for over 70 days since March 5 BC, and that would have been first seen by the Magi rising in the East in the morning sky, prompted the Magi to set out to Palestine in March 5 BC.
In this theory the Magi would have told Herod in December 6 BC (or in April/May 5 BC in the Humphreys 1995 version) that they had first seen the star (Jupiter) "at its rising" in the first conjunction of May 7 BC, and the 2-year range for the age of infants to be killed in the Bethlehem area would have been a direct consequence of that information.
1 Ferrari-D'Occhieppo, Konradin (1989). "The Star of the Magi and Babylonian Astronomy" in "Chronos, kairos, Christos: Nativity and chronological studies presented to Jack Finegan" edited by Jerry Vardaman and Edwin M. Yamauchi
2 Parpola, Simo (2001). "The Magi and the Star. Babylonian Astronomy Dates Jesus' Birth". Bible Review, December 2001, p. 16-23, and p. 52-54. Online as chapter 2 (pp. 13(20) - 24(31)) of the e-book "The First Christmas. The Story of Jesus' Birth in History and Tradition"
3 Molnar, Michael R. (1999). "The star of Bethlehem: the legacy of the Magi"
Michael R. Molnar site: http://www.eclipse.net/~molnar/
4 M.M. Dworetsky and S.J. Fossey (1997). "Lunar Occultations of Jupiter and Saturn, and the Star of Bethlehem" Originally published in The Observatory, Vol. 118, No. 1142, pp. 22-24, 1998 February.
Points out that both lunar occultations of Jupiter were preceded by lunar occultations of Saturn in the constellation of Pisces one day before.
5 Humphreys, Colin J. (1995). "The Star of Bethlehem". Science and Christian Belief , Vol 5, (October 1995): 83-101.