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At full moon on 21 January, there is a total eclipse of the moon, visible in Europe in the early hours, and also in the Americas. The penumbral phase begins at 02:36 (UTC); this is when the first area on the moon has the first small amount of light blocked by the Earth, and is hard to see; the partial phase starts at 03:33, and is when the first area of the moon has its direct sunlight completely blocked by the Earth, and you can normally see some darkening before this happens, as more and more sunlight is blocked from the leading edge of the moon. Totality lasts from 04:41 to 05:43, and this is when the whole of the moon has all of its direct light blocked; however, some red light still reaches the moon through the Earth’s atmosphere. If you were standing on the moon, you’d see a fiery red ring around the Earth. The partial phase ends at 06:50, and the penumbral phase at 07:48, just before moonset in SE parts of Britain.
Mars is in the evening sky, and sets in the late evening. The crescent moon is nearby on 12 January.
Mercury may still be just visible in the dawn at the start of the month, especially as seen from the southern hemisphere, before it moves towards the far side of the sun. Further up in the eastern sky is Jupiter, rising about two and a half hours before the sun, and shining brightly in the pre-dawn sky. Further still out from the dawn is Venus, which reaches 47°W of the sun on 6 January, and shines very brightly and prominently during the last part of the night. As the month progresses, Venus approaches the less bright Jupiter, and the two will look very well together in the east before it starts getting light. On the morning of 22 January, Venus passes north of Jupiter, and the moon is between Jupiter and Venus on the morning of 31 January.
By the end of the month, Saturn makes its way from behind the sun into the eastern morning sky, and you should be able to see the three planets in a curving line; in order out from the dawn, they are Saturn, then Venus, then Jupiter. On the morning of 2 February, the moon is very close to Saturn, and occults it in the dark as seen from parts of Western Europe and the Maghreb, but the moon is a very old crescent, and near to the sunrise, so there are few places where the occultation can be seen well. The only part of Britain & Ireland where the occultation can be seen at all is South-East England. From London, the occultation starts well before moonrise, and ends at 06:33 (UTC) when the moon and Saturn are still very near the SE horizon. When the occultation ends in Paris at 07:32 (06:32 UTC), the moon and Saturn are a little higher in the SE, although the twilight will be somewhat brighter. In Barcelona, the occultation also starts before moonrise, and finishes at 07:27 (06:27 UTC), before the start of civil twilight. Here you may well be able to see Saturn emerge from behind the dark edge of the moon. In both Rome and Tunis, the occultation starts when the moon and Saturn are high enough above the horizon to be seen fairly easily, but ends in daylight. Saturn begins to be covered by the lit edge of the moon in Tunis at 06:18 (05:18 UTC) and in Rome at 06:28 (05:28 UTC).
Mars is in the evening sky, setting in the late evening. The crescent moon is nearby on 10 February. Also nearby is the ice giant Uranus. Mars passes into the constellation Aries on 13 February, and the same day is a degree north of Uranus. Look for the pair in binoculars, as Uranus is just too faint to see with the naked eye. From Europe, look for them together on the evenings of 12 and 13 February.
On 27 February, Mercury reaches 18°E of the sun in the western evening sky. It’s not a good view from the southern hemisphere, but from the north, for about a week either side of the date, you may be able to see Mercury low down in the west as evening civil twilight ends.
Saturn is low in the eastern morning sky in the constellation Sagittarius. Much brighter, and further out from the dawn is Venus, the bright morning star; at the start of the month it is approximately half-way between Saturn and Jupiter, but quickly moves towards Saturn, and passes north of it on 18 February. Jupiter rises further from the dawn (about four hours in advance of the sun, mid-month), and shines brightly, through not as brilliantly as Venus. The moon is near Jupiter on the mornings of 27 and 28 February. On the morning of 2 March, as seen from Europe, the moon is between Venus and Saturn. Indeed, the moon occults Saturn in the dark as seen from central parts of the Pacific early on 2 March (local time to the west of the date line). From Tarawa Atoll in Kiribati, the bright side of the moon covers Saturn at 04:53 on 2 March (1 Mar, 16:53 UTC), and Saturn emerges again from the dark side at 05:27 (17:27 UTC). On 3 March, the old moon is between Venus and the dawn, though somewhat to the south of Venus.