Constellations of the Third Age
I'll try to explain the star map published in Reunion 15. Any person habituated to see star maps will soon realise that there are important irregularities in the distribution of the stars. That's because the presented map belongs to the Second Age, before the Downfall of Númenor, in a time when world was flat and stars were stationary in Ilmen (see the Ambarkanta, in The Shaping of Middle-earth), and only those that we know as planets moved from East to West, like the Sun and the Moon.
So the six planets (see C Tolkien's introductory words to the Index of Morgoth's Ring) and Eärendil are not represented. The map represents what could be seen from the centre of the flat world (more or less Khand to the SE of Mordor), and so it has a round shape, with some eccentricity. Three exact data, given by Tolkien, allow us to "bind" the dome of stars.
The Valacirca (seven main stars of the Ursa Major) is said to be placed by Varda as a menace for Morgoth in the North. Soronúmë is the "Eagle of the West", and Helluin (Sirius) was the star that awoke the Elves, so it has to be over the regions surrounding Cuiviénen, the "Waters of Awakening".
Apart from these three points mentioned in The Silmarillion, we have too the reference from the Lost Tales that Morwinyon (Arcturus) is a sparkle in the West left by Varda in her coming back to Valinor after the second kindling of the stars.
But this can hardly agree with what is said about Soronúmë. So now, having explained some features of the map, I'll tell the story of the establishment of these constellations of the ancient times (there's more information in chapter 3 of Quenta Silmarillion).
Though the first stars were placed in Ilmen by Varda without any order, many of those that awoke the Elves were placed deliberately making figures, but Elves and Men recognised these shapes and many other.
At first, there were six moving stars, those that were later called planets, named by the Elves already at Cuivienen: Carnil (Mars), Luinil (Neptune), Nénar (Uranus), Lumbar (Saturn), Alcarinquë (Jupiter), and Elemmírë (Mercury).
To those was later added Eärendil, whose Silmaril shined more than any star. Helluin, the brightest star of the sky, awoke the Elves; they soon turned their look westward, because the East was poor in stars, and they identified Anarríma, Wilwarin and, of course, Valacirca, the powerful defiance of Varda against Melkor. Menelmacar the Elves included in Anarríma until in Valinor they knew of its true signification.
But, while the Great Journey advanced, the Eldar gave names to other groups of stars: Morwinyon - the only great star, with Helluin, not included in a constellation; Telumendil, "Guide of the Sky" - named when the Elves turned to the South-west after leaving the shores of the Sea of Helcar; Soronúmë - remembering the first eagle that the Elves saw, going into Eriador; Helqua, "Swan of Ice" - for the birds of the Teleri, identified by the Vanyar (creators of the constellation) with ice; and Ilmarok, dedicated to the horses of Oromë.
Later, in Eldamar, the Elves created other constellations dedicated to different maiar: Oaritse - for the Oarni and the mermaids servants of Uinen; Tanrínye - for Armaiti, the main female smith and jeweller servant of Aulë; and Telimektar - servant of Tulkas. With these thirteen constellations the Noldor fled from Valinor.
Just arrived in Beleriand, after the death of Fëanor, the Noldor created the constellations of Fëanor - in the extreme West, over Valinor - and of his father Finwë, next to the Valacirca. In the same way, they created too, after Fingolfin's death, the constellation of Ringil, Fingolfin's sword.
But after the terrible defeat of the Nirnaeth Arnoediad, the Noldor only could see ominous symbols at the North: two dragons in the horizon and one in the centre of the sky, menacing some "fair" Varda's constellations. Only after the War of Wrath they were identified as Ringalóke, Morlóke and Urulóke, these two with wings. The Númenóreans added four constellations more, reuniting the remnant stars of the flat world. Vardamir Nólimon introduced the Velike Makar Vanya remembering the Vanyar warriors of the War of Wrath, whose memory was still strong among Númenóreans.
Uin was introduced by Tar-Meneldur, in a time when Númenóreans were already a people devoted to the sea. Raumal and Aiwanar were created by the astronomers of Oromet when Tar-Ciryatan took the Sceptre, because this King wanted to celebrate in the skies the beauty of other parts of the world.
Though Raumal had been known until then by the Elves as Moth, "the Sheep", everyone, both in Endor and Aman, accepted the change. That's why, among the three constellations introduced by Númenóreans, only Raumal was placed to the North: the other three were to the South-east, where Elves because of the distance, had not given importance to their stars.
When the World was made round, new stars were revealed in the southern skies as the last gift from Varda to the mortal lands. Among the ancient constellations, only those placed near the horizons changed a little their shapes, though they kept being recognisable.
Some of them, because of their evident shape, have kept an approximate meaning until our days: Ringalóke (Draco), Menelmacar (Orion), Velike Makar Vanya (Perseus), Uin (Cetus, the Whale), Ilmarok (Pegasus), Urulóke (Hydra), Raumal (the Lion), the crown of Tanrínye (Corona Borealis), Helqua (Cygnus, the Swan) and Soronúmë (the Eagle).
But Aiwanar (the Phoenix), recuperated in 1603 AD by J Bayer, was suggested to this astronomer by Menelnol, a Vanya sent to be his assistant for some time by Maglor; though Bayer, of course, didn't know the true identity of his collaborator.