I How to calculate an unknown planet's orbital period?

[NODARman]

In 1846 three astronomers and mathematicians discovered Neptune because Uranus wasn't quite moving as Newton's law of gravity explains. So they did calculations and point the telescope at a specific part of the sky. They discovered Neptune. What formulas did they use? How did they calculate this kind of hard problem?

 

[Astronuc]

On the night of Sept. 23-24, 1846, astronomers discovered Neptune, the eighth planet orbiting around the Sun. The discovery was made based on mathematical calculations of its predicted position due to observed perturbations in the orbit of the planet Uranus. The discovery was made using a telescope since Neptune is too faint to be visible to the naked eye, owing to its great distance from the Sun. Astronomers soon discovered a moon orbiting Neptune, but it took more than a century to discover a second one. Our knowledge of distant Neptune greatly increased from the scientific observations made during Voyager 2’s flyby in 1989, including the discovery of five additional moons and confirmation of dark rings orbiting the planet.

https://www.nasa.gov/feature/175-years-ago-astronomers-discover-neptune-the-eighth-planet

With the 1781 discovery of Uranus, the number of known planets in the solar system grew to seven. As astronomers continued to observe the newly discovered planet, they noticed irregularities in its orbit that Newton’s law of universal gravitation could not fully explain. However, effects from the gravity of a more distant planet could explain these perturbances. By 1845, Uranus had completed nearly one full revolution around the Sun and astronomers Urbain Jean-Joseph Le Verrier in Paris and John Couch Adams in Cambridge, England, independently calculated the location of this postulated planet. Based on Le Verrier’s calculations, on the night of Sept. 23-24, 1846, astronomer Johann Gottfried Galle used the Fraunhofer telescope at the Berlin Observatory and made the first observations of the new planet, only 1 degree from its calculated position. In retrospect, following its formal discovery, it turned out that several astronomers, starting with Galileo Galilei in 1612, had observed Neptune too, but because of its slow motion relative to the background stars. did not recognize it as a planet.

I would expect that the perturbation calculations have been published.

For a modern example, https://aapt.scitation.org/doi/abs/10.1119/1.16307?journalCode=ajp

http://www.pas.rochester.edu/~blackman/ast104/perturbations.html

So, for an unknown planet, one would need a 'known' planet with which to measure a perturbation, or a high resolution telescope with which to compare the 'unknown' planet against the stellar background.

When Clyde Tombaugh discovered Pluto in 1930 using the 13-inch telescope at Lowell Observatory, it was only a point of light, detected among the background stars by its extremely slow motion. That motion translated to a 248-year orbital period, placing it at the edge of the solar system. It was a fantastic discovery, but Pluto at that time was not recognized as a new class of object, nor could it be, without knowing its mass. The mass, size and density of Pluto were for decades considered to be similar to the planet Mars.

https://www.nasa.gov/exploration/whyweexplore/Why_We_23.html

historically, the Pluto situation has occurred before in the solar system. Two centuries ago, in 1806, William Herschel’s discovery of the 7th planet, Uranus, was exactly 25 years in the past. But astronomers were rejoicing in the discovery of 3 new planets in the last three years, Ceres in 1801, Pallas in 1802 and Juno in 1804. And Vesta was about to be discovered in 1807. So in 1806, astronomers thought there were 11 planets. Astronomer James Hilton has shown how for almost 50 years the Nautical Almanacs listed 12 planets, including Vesta. Then, 39 years after those 4 new planets had been discovered, came a problem: in 1847 three new one were found, and by the end of 1851 there were 15. Only by the mid-19th century, ‘once their numbers grew too large to fit the existing scheme of classification,’ were ‘minor planets’ or ‘asteroids’ accepted as a class of their own. . . .