In short, I am asking about the range of year lengths for a planet orbiting in the circumstellar habitable zone of 40 Eridania A.
Since Vulcan, one of the most famous planets in science fiction, is usually supposed to orbit 40 Eridani, I am also asking how the orbit of Vulcan around 40 Eridania A could fill all the fictional requirments, including having a year length close to one Earth year.
Part One: The Star of Vulcan.
One of the most famous fictional planets is Vulcan in Star Trek.
In the Enterprise episode "home". 22 Oct. 2004, T'pol and Tucker travel from Earth to Vulcan and later:
T'POL: I'm sorry.
TUCKER: You're sorry. You brought me sixteen light years just to watch you get married to someone you barely know.
In the Enterprise episode "Daedalus", 14 Jam. 2005, a hypothetical long range transporter system is discussed:
EMORY: Sub-quantum teleportation. You step on to a transporter on Earth, a few seconds later, you're on Vulcan.
TUCKER: That's over sixteen light years.
Those two episode seem to confirm that Vulcan is between sixteen and seventeen light years from Earth. fortunately or unfortunately, there are about a dozen known star systems which are between sixteen and seventeen light years from Earth.
They include a few well known stars such as Altair, 70 Ophiuchi, and 40 Eridani, and Star Trek fans often think 40 Eridani is the star of Vulcan.
Part Two: 40 Eridani.
The system of 40 Eridani consists of three stars.
40 Eridani A is a main-sequence dwarf of spectral type K1, 40 Eridani B is a 9th magnitude white dwarf of spectral type DA4, and 40 Eridani C is an 11th magnitude red dwarf flare star of spectral type M4.5e. When component B was a main-sequence star, it is thought to have been the most massive member of the system, but ejected most of its mass before it became a white dwarf. B and C orbit each other approximately 400 AU from the primary star, A. Their orbit has a semimajor axis of 35 AU (which is the approximate average distance between B and C) and is rather elliptical (eccentricity 0.410).
The habitable zone of 40 Eridani A, where a planet could exist with liquid water, is near 0.68 AU from A. At this distance a planet would complete a revolution in 223 Earth days (according to the third of Kepler's laws) and 40 Eridani A would appear nearly 20%[note 3] wider than the Sun does on Earth. An observer on a planet in the 40 Eridani A system would see the B/C pair as unusually bright (magnitudes -8 and -6) white and reddish-orange stars in the night sky.
It is unlikely that habitable planets exist around 40 Eridani B because they would have been sterilized by its evolution into a white dwarf. As for 40 Eridani C, it is prone to flares, which cause large momentary increases in the emission of X-rays as well as visible light. This would be lethal to Earth-type life on planets near the flare star.
40 Eridani A or Keid has one known planet, 40 Eridani A b, which has about eight to nine times the mass of Earth, and orbits with a semi-major axis of 0.224 AU, far closer to the star than the inner edge of the habitable zone. It is possible that the perturbing effects of that planet on any hypothetical planets in the circumstellar habitable zone of 40 Eridani A would be important.
Until the hypothetical future time when astronomers might hypthetically prove that there aren't any habitable-sized planets orbiting in the circumstellar habitable zone of 40 Eridani A, it will probably be the most favored candidate for the star of Vulcan.
The length of the orbital period or year of a hypothetical planet in the circumstellar habitable zone of 40 Eridani A depends on its exact orbital size and on the the mass of 40 eridania A, which is about 0.84 times the mass of the Sun.
The inner and outer edges of the circumstellar habitable zone of 40 Eridani A are determined by the luminosity of 40 Eridani A, which is 0.457 plus or minus 0.002 that of the Sun.
Part Three: Habitable Zones.
However, there is little certainty about the inner and outer edges of the circumstellar habitable zone of the Sun.
Here is a link to a list of 15 diferent estimates and calculations of the inner and outer edges, or both, of the circumstellar habitable zone of the Sun made in the last 50 years:
Note that they differ greatly.
I see that some of them extend the inner and outer edges of the Sun's habitable zone only for planets with atmospheric and/or other conditins very different from those on Earth. Unprotected humasn would not be able to live on such planets - unprotected humans can't survive even in many parts of Earth's biosphere.
The only estimate, as far as I know, of the limits of the Sun's circumstellar habitable zone for planets habitable specifically for humans and for animals with the same environmental requirements, instead of the habitable zone for planets habitable for Earth life in general, is that of Stephen H. Dole in Habitable Planets for Man, 1964, 2007.
Part Four: The Vulcan Climate.
And of course Vulcan is supposed to be habitable, though rather uncomfortable, for humans, since human characters have been depicted on Vulcan.
in "Amok Tine":
KIRK: It's lovely. I wish the breeze were cooler.
MCCOY: Yeah. Hot as Vulcan. Now I understand what that phrase means.
KIRK: The atmosphere is thinner than Earth.
MCCOY: In this climate? If the heat doesn't get you, the thin air will. You can't do it!
MCCOY: Is this Vulcan chivalry? The air's too hot and thin for Kirk. He's not used to it.
So the heat of Vulcan indicates that Vulcan might be close enough to 40 Eridania A to receive significantly more radiation than Earth receives from the Sun, thus making the length of the Vulcan orbital period and year even shorter than if Vulcan orbited at the distance to receive the same amount of radiation as Earth gets.
But there is a complication.
Part Five: the Possible Need for a Vulcan Year About as Long as Earth's.
Since I am working on Star Trek chronology, and may publish something on the subject someday, that may count as repairing or rebuilding someone else's fictional universe or building my own fictional universe based on the Star Trek fictional universe. Does nonficiton about fiction count as nonfiction or as fiction for the purpose of Worldbuilding Stack Exchange?
My answer to:
Includes a section which uses data from the animated Star Trek episode "Yesteryear" to indicate that Spock was aged about thirty two to thirty seven during the five year mission in TOS.
But at least one of the ages and time spans mentioned was in Vulcan years. So Spock's age range of 32 to 37 during TOS might be in Vulcan years, which could be considerably shorter than Earth years if Vulcan orbits 40 Eridania A. Making Spock significantly younger than 32 to 37 Earth years during TOS could eliminated the problem that Spock seems a bit too old to have a mother as young as Amanda, but would create a problem with how long (presumably in Earth years) Spock is said invarius epsiodes to have served in Starfleet.
If the Vulcan year length is within about 10 percent of the legnth of an Earth year, Spock's age (posiblyin Vulcan years) of 32 to 37 in TOS and TAS according to "Yesteryear" would range from about 28 to 34 Earth years on the low end up to about 35 to 40 Earth years on the high end.
Such comparatively minor deviations from having Spock's age in "Yesteryear" be in Earth years would produce comparatively minor problems in fitting his biography into Star Trek chronology.
But if a Vulcan year was about three quarters or 0.75 of an Earth year long, about 280 Earth days) Spock's age in TOS and TAS would be about 24 to 28 Earth years, and therw would be really big problems fitting Spock's biography into biography into Star Trek chronology.
According to Wikipedia, a planet in the habitable zone of 40 Eridania would be about 0.68 AU from the star and have a day about 223 Earth days long.
So that could make Spock about 20 to 23 years old, which really mess up the Star Trek chronology. And Vulcan is said to be hotter than earth, so its year should be somewhat shorter, rather than somewhat longer, than 223 Earth days.
Some Star Trek fans don't count the animated Star Trek as canon, and so have no problem with its evidence of Spock's age.
Other fans do count it as canon, and thsu may have a problem with Spock's age if Vulcan years are significantly shorter than Earth years. Those fans may have to give up thinking that Vulcan is a planet of 40 Eridani, even though there aren't many better choices among the few stars between 16 and 17 light years from Earth, and even though 40 Eridani has long been considered to be Vulcan's star.
I would appreciate it if someone could calculate the range of year length's for a planet orbiting between the inner and outer edges of 40 Eridani A's circumstellar habitable zone. And state which estimate of the Sun's circumstellar habitable Zone they used as a basis.
And I would also appreciate it if someone could figure out a way to have a planet orbit 40 Eridania, be hotter than Earth, and have the length of its year be approximately equal to that of an Earth year.