Tolkien does write that Middle Earth is essentially our earth, in a fictional past:
I have, I suppose, constructed an imaginary time, but kept my feet on my own mother-earth for place. I prefer that to the contemporary mode of seeking my own remote globes in 'space'. However curious, they are alien, and not lovable with the love of blood-kin. [...] Many reviewers seem to assume that Middle-earth is another planet!
(Letters of J.R.R. Tolkien, letter No. 211 (a very good letter))
Many of the ME stars are also references to real stars. Of course, Arda changes radically after the first age. Before that, Aman and Middle Earth were two continents on a flat(!) Arda, encircled by the Sun (Anar) and the Moon (Rana). Then, Illúvatar curved down the corners of the Earth (or something), and made Arda round. And after the fall of Númenor, Illúvatar removed Aman from Arda, and changed continents and oceans.
So, to answer your question: third age (and later) Arda is practically identical to Earth (astronomically), and first age somehow seems to be flat and different. (Yes, I know, it's weird)
As a reference: This answer of mine includes a map of Arda 'after the war of the Gods' (first age)
EDIT: here is the 'changing of the world' bit from Akallabêth:
Then Manwe upon the Mountain called upon Illúvatar, and for that time the Valar laid down their government of Arda. But Illúvatar showed forth his power, and he changed the fashion of the world; and a great chasm opened in the sea between Númenor and the Deathless Lands, and the waters flowed down into it, and the noise and smoke of the cataracts went up to heaven, and the world was shaken. [...]
But the land of Aman and Eressea of the Eldar were taken away and removed beyond the reach of Men for ever. [...]
For Illúvatar cast back the Great Seas west of Middle-earth, and the Empty Lands east of it, and new lands and new seas were made; and the world was diminished, for Valinor and Eressea were taken from it into the realm of hidden things.
And here is a bit from 'Of the Sun and the Moon and the Hiding of Valinor' (Silmarillion):
But the Flower and the Fruit [of the now dead Two Trees] Yavanna gave to Aule, and Manwe hallowed them, and Aule and his people made vessels to hold them and preserve their radiance: as is said in the Narsilion, the Song of the Sun and the Moon. These vessels the Valar gave to Varda, that they might become lamps of heaven, outshining the ancient stars, being nearer to Arda; and she gave them power to traverse the lower regions of Ilmen, and set them to voyage upon appointed courses above the girdle of the Earth from the West unto the East and to return.
And to show that the night sky was indeed the same as ours:
In 'Of the Coming of the Elves and the Captivity of Melkor', Varda makes the stars (and planets, all are described as stars here):
[...] and therewith she made new stars and brighter against the coming of the First-born; wherefore she whose name out of the deeps of time and the labours of Eä was Tintalle, the Kindler, was called after by the Elves Elentári, Queen of the Stars. Carnil and Luinil [lit. red and blue (stars?)], Nenar and Lumbar, Alcarinque and Elemmire she wrought in that time
Those were the planets
and many other of the ancient stars she gathered together and set as signs in the heavens of Arda: Wilwarin, Telumendil, Soronúme, and Arríma; and Menelmacar with his shining belt, that forebodes the Last Battle that shall be at the end of days. And high in the north as a challenge to Melkor she set the crown of seven mighty stars to swing, Valacirca, the Sickle of the Valar and sign of doom.
(This sickle is also found in LotR, chapter 'Strider'):
The Sickle* was swinging bright above the shoulders of Bree-hill.
and the footnote:
*The Hobbits' name for the Plough or Great Bear
More on the planets:
There are some notes on Star-names at the end of "Morgoth's Ring":
In the second, all but final, draft my father is seen in the act of devising the names of the constellations, with various experimental forms before those that appear in the final text were reached; but he set down the names of the stars without hesitation, thus: *Karnil, Luinil, Nénarm Lumbar, Alkarinque, Elemmire. Above Karnil he wrote 'M', above Lumbar 'S', above Alkarinque 'Jup', and above Elemmire again 'M'. No letter stands above Luinil, but above Nénar there is an 'N' which was struck out.
Now if Alkarinque is Jupiter, then a great red star named Karnil and marked with an 'M' must be Mars - which in turn leads to the identification of Lumbar ('S') with Saturn, and Ellemire ('M') with Mercury. [...]
He is unsure about the the identities of Luinil and Nénar. They seemed to be Neptune and Uranus, but that doesn't fit in with the text 'new stars and brighter against the coming of the First-born'
So we know:
- Carnil is Mars
- Luinil is not definite, 'luin' is blue, so people guess it might be Neptune, or even Orion (a constellation)
- Nénar is also not definite, maybe Uranus
- Lumbar is probably Saturn
- Alcarinque is Jupiter
- Elemmire is Mercury