The first exoplanets were detected in 1992 and since then nearly 5000 been detected. The pace of these discoveries will accelerate greatly once the James Webb Space Telescope (JWST) becomes operational later this year (as of the time of this writing, the JWST was just three days from reaching its final position at the Lagrange Point L2. (Related post from book with profile of 20 exoplanets).
The site Big Think has a nice estimate of planets in the universe. It estimates that there are over a trillion galaxies and each galaxy has a trillion to ten trillion planets orbiting its starts putting the total at 1025. The article brings up the fact that there are also rogue planets that do not orbit starts which can put the total number of planets 10 to 10,000 times higher. These rogue planets are also described in the book, Imagined Life: Math Vacation: Book Review: Imagined Life by James Trefil and Michael Summers (jamesmacmath.blogspot.com).
Also see this post regarding number of black holes in the universe: Math Vacation: How Many Black Holes are there in the Universe? (jamesmacmath.blogspot.com)
Update 1/24/2022: The JWST successfully entered into its halo orbit at L2 - Where Is Webb? NASA/Webb.
Update 2/12/2022: The JWST took its first image of a star. This part of the long process to align the 18 primary mirrors and the secondary mirror to produce a single image. The image (below) and the alignment process are described at the NASA JWST site.
Image Credit: NASA - Post: Edit (blogger.com)
Update 2/22/2022
The alignment of the JWST mirrors continues as the image above has been adjusted to the hexagonal arrangement of the individual mirrors: JWST alignment continues, light-points now in hexagonal formation - Space Explored
The alignment of the JWST mirrors continues as the image above has been adjusted to the hexagonal arrangement of the individual mirrors: JWST alignment continues, light-points now in hexagonal formation - Space Explored
Furthermore, considering a prior post about other civilizations in our galaxy, the JWST may be able to gain evidence of remote civilizations by detecting their pollution or by detecting evidence of photosynthesis on distant planets: Could Nasa's James Webb Space Telescope detect alien life? - BBC News.
Update 2/26/2022
The JWST made another step toward its mirror alignment. It stacked the eighteen individual images into one. Before and after images below (from Webb Mirror Alignment Continues Successfully – James Webb Space Telescope (nasa.gov)).
The specialty of the JWST is imaging in the infrared spectrum. NASA has a good article here: https://webb.nasa.gov/content/science/firstLight.html.
The James Webb Space Telescope recently found some rogue planets:
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