What are the upper and lower year lengths for a habitable planet of 40 Eridani A?

Question

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.

https://en.wikipedia.org/wiki/List_of_star_systems_within_16%E2%80%9320_light-years[1]

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.[citation needed] B and C orbit each other approximately 400 AU from the primary star, A.[13] 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).[12]

https://en.wikipedia.org/wiki/40_Eridani#Properties[2]

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.[13]

https://en.wikipedia.org/wiki/40_Eridani#Potential_for_life[3]

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:

https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates[4]

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.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[5]

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.

And:

MCCOY: In this climate? If the heat doesn't get you, the thin air will. You can't do it!

And:

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:

https://scifi.stackexchange.com/questions/240373/how-old-was-spock-in-star-trek-while-he-was-serving-on-the-enterprise/240425#240425[6]

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.

https://en.wikipedia.org/wiki/40_Eridani#Potential_for_life[3]

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.

Answer 1

I have thought of a partial answer to the part of my question about giving a habitable planet, with at least the same average temperature as Earth, in the 40 Eridani system (or any star system where the star is signicantly less luminous than the Sun) a year approximately as long as an Earth year, within about ten percent or so.

Even though the internal heat of Earth does not provide signicant heat to the surface of Earth, it is possible that some other planets, moons and other worlds might have enough internal heat to raise their surface temperatures far above the temperatures resulting from the radiation they receive from their stars. Such worlds could be as warm as Earth at much greater distances from their stars than the distances where they would receive as much radiation from their stars as Earth gets from the Sun.

One of the sources of internal heat that could be greatly increased is tidal heating. Theoretically, the gravitational forces of every object in a solar system exerts a tidal force on every other object in that solar system. Thus the interior of an object is constantly flexed and distorted to varying degrees by the pull in various directions of all the other objects in the system, resulting in the release of varying abounts of heat.

An object which is large enough and/or close enough can produce strong tides and significant internal heating in another object. One object very large and close will tend to put the second object into a nearly circlar orbit around it and minimize tidal forces. But two or more objects with different orbits exerting strong tidal forces in different directions on an object can keep on exerting strong tidal forces on it for long periods of time.

In "Exomoon Habitability Constrained by Illumination and Tidal Heating", Rene Heller and Roy Barnes, Astrobiology, volume 13, number 1, 2013, many factors affecting the potential habitability of hypothetical giant exomoons orbiting giant exoplanets in other star systems are discussed.

https://faculty.washington.edu/rkb9/publications/hb13.pdf[1]

They consider the amount of tidal heating an exommon has to be a potentially important factor in possibly making an otherwise too cold moon warm enough for liquid water and life, or in possibly making an otherwie warm enough moon too hot so that it suffers a runaway greenhouse effect like the planet Venus and turns too hot for life.

In section Four, Orbits of Habitable Exomoons, they say:

By analogy with the circumstellar habitable zone for planets, we can imagine a minimum orbital separation between a planet and a moon to let the satellite be habitable. The range of orbits for habitable moons has no outer edge, except that Hill stability must be ensured. Consequently, habitability of moons is only constrained by the inner edge of a circumplanetary habitable zone, which we call the ‘‘habitable edge.’’ Moons inside the habitable edge are in danger of running into a greenhouse by stellar and planetary illumination and/or tidal heating. Satellites outside the habitable edge with their host planet in the circumstellar IHZ are habitable by definition.

So an otherwise habitable exomoon orbiting too clos to its giant exoplanet, closer than the habitable edge, could overheat and become uninhabitable from excessive tidal heating.

This indicates that in some situations tidal heating of a world by a nearby world could make it warm enough for life despite not receiving sufficient illumination from its star.

So a planet with temperatures suitable for life can orbit outside the circumstellar habitable zone of its star and have an orbital period longer than the orbital period it would have if it orbited in the circumstellar habitable zone of its star - if the planet has another source of heat, such as tidal heating from other worlds.

Thus some writers who are informed that they have given their fictional planet a year too long to fit inside the habitable zone of its star may be able to get out of the corner they wrote themselves into by supposing that the planet has strong tidal heating from nearby worlds - if they haven't already eliminated the possibility of other worlds close enough to that planet to give it strong eneough tidal heating.

Adn thus the planet Vulcan could have a year about one Earth year long and be warm enough for life, even a bit hotter than Earth, if it had sufficient tidal heating from other worlds to make up for the much less illumination it would receive from its star.

I found statement that a world orbiting 40 Eridani A at a distance of about 0.9 AU would have a year 355 days long, close enough to an Earth year, and get about as much illumination, and thus heat, from 40 Eridani A as Mars gets from the Sun.

Gravitational force decreases with the square of the distance, I think. If a planet was 0.9 AU from from the Sun, the tidal force of the Sun would be about 1.243 times as strong as the tidal force on the Sun on Earth. Since 40 Eridani A has 0.84 the mass of the Sun, its gravitational force on a planet orbiting at 0.9 AU would be about 1.0368 that of the Sun on Earth.

One planet has been discovered so far in the 40 Eridani system, 40 Eridani A b, with a mass of 8.47 plus or minus 0.47 that of the Earth -8.0 to 8.94 times that of Earth. It orbits with a period of 42.378 Earth days with a semi-major axis of 0.2244 AU.

When at its closest, lined up directly between 40 Eridani A and Vulcan, in what is called inferior conjunction, 40 Eridania A b would be 0.67 AU from Vulcan. When at its farthest, lined up directly beyond 40 Eridani A from Vulcan, in what is called superior conjunction, 40 Eridania A b would be 1.12 Au from Vulcan.

At other times the distance would be tween those limits and the star and the inner planet would be in slightly different directions as seen from Vulcan.

Since 40 Eridani A b has 8.00 to 8.94 the mass of Earth, and the Sun has a mass of about 333,000 Earth mass, 40 Eridani A b has about 0.000024 to 0.000026 times the mass of the Sun. So my calculations indicate it should have between 0.0000189 and 0.0000416 times the gravitational force on Vulcan as the Sun has on Earth, thus that planet's gravitational pull on Vulcan would be very small compared with the gravitational force of the Moon on Earth.

So at least one additional object, massive and/or close to the hypothetical orbit of Vulcan, must be imagined to exist in the 40 Eridani A system if Vulcan is hot and also has a year similar in length to an Earth year.

The candidates would be an object orbiting in the same orbit as Vulcan, and another planet with a different orbit than Vulcan.

An object sharing tthe same orbit as Vulcan would be either a natural satellite orbiting Vulcan, or the other member of a double planet that included Vulcan, or a giant planet which Vulcan was a natural satellite or moon of.

Of course Spock said that "Vulcan has no moon", so that seems to eliminate a moon of Vulcan as the source of Vulcan's hypothetical tidal heating. That leaves the possibility of Vulcan having a twin planet or being a giant moon of a planet much larger than Vulcan.

And of course it would be possible for Vulcan to periodically pass close to another planet orbiting 40 ERidani A and perhaps having significant tidal heating.

One problem with tidal heating of Vulcan is that only one planet has been detected orbiting 40 ERidani A, and that planet is much smaller and less massive than giant planets like Jupiter. It is possible that the studies of 40 Eridani A would have discovered any giant planets in the system, no matter where they orbited.

The answer to this question:

https://astronomy.stackexchange.com/questions/27697/low-mass-cut-off-for-the-dharma-planet-surveys-detection-of-habitable-planets-a[2]

Indicates that a planet the size of Earth cannot be detected with present instruments in the 40 Eridani A system orbiting at the inner edge of the circumstellar habitable zone, and would be even less easy to detect if it was orbiting farther out. Thus Earth mass and larger planets could be orbiting 40 Eridani A in the habitable zone and beyond it and not have been detected yet. But I don't know how many times more massive than Earth they could be and not be detected yet.

So using tidal heating from neaby astronomical objects would be one possible way to design a habitable planet a bit outside the habitable zone of its star in general, and a planet Vulcan in the 40 Eridani A system with a year similar in length to an Earth year, in particular.

Answer 2

In case anyone wants to try designing a solar system where a habitable planet orbits a star with a mass similar to that of 40 Eridania A and yet has a year that needs to be similar in length to an Earth year, here is the evidence.

According to my calculations from "Yesteryear" explained below, a Vulcan year should either equal 0.923 (zero point nine two three) to 1.270 (one point two seven zero) Earth years or equal 0.967 (zero point nine six seven) to 1.550 (one point five five zero) Earth years.

Calculating from a Julian Calendar year 365.25 (three hundred sixty five point two five) Earth days long, a Vulcan year should either be 337.125 (three hudnred thirty seven point one two five) to 463.966 (four hundred sixty three point nine six six) Earth days, or 353.452 (three hundred fifty three point four five two) to 566.137 (five hundred sixty six point one three seven) Earth days.

According to Wikipedia:

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.

https://en.wikipedia.org/wiki/40_Eridani#Potential_for_life[1]

Vulcan is supposed to be hot, so if anything it should receive more radiation from 40 Eridani A than Earth gets from the Sun, and so should be a little closer than 0.68 (zero point six eight) Astronomical Units from 40 Eridani A, and so have a year shorter than 223 (two hundred and twenty three) Earth days.

A year 337.125 to 463.966 Earth days long would be about 1.511 (one point five one one) to 2.080 (two point zero eight zero) times as long as 223 Earth days.

A year 353.452 to 566.137 Earth days long would be about 1.5849 (one point eight four nine) to 2.538 (two point five three eight) times as long as 223 Earth days.

Planets orbiting that much farther from 40 Eridania should have temperatures as cold as Mars instead of being hot compared to Earth.

But a hypothetical planet orbiting that far from 40 Eridani A could be at least as hot as Earth, if the heat from the star's radiation was supplimented by a vast amount of internal heat, possibly resulting from tidal heating caused by tidal interactions with companion worlds of some type.

Thus designing one or more companion planetary mass objects for Vulcan, with the correct masses and distances to produce extensive tidal heating to provide probably more tidal heat than the heat Vulcan receives from its star, would make it far more believable that Vulcan could both have a year as long as "yesteryear" indicates and alo orbit around 40 Eridani A.

The evidence for calculating the year length of Vulcan.

In "Yesteryear" Kirk and Spock return from a journey to the past to find that Spock has been erased from history.

KIRK: I don't know what's going on, but the first officer of this ship will be treated with respect.

(An Andorian enters)

THELIN: Captain, I assure you no one has ever treated me otherwise.

KIRK: Who are you?

MCCOY: Oh, I thought sure you'd know Thelin by now, Jim. He's been your first officer for five years.

If Thelin became first officer to Kirk at the same time as Spock did in the "prime" universe, Spock should have done so about five years, more or less, before "Yesteryear".

Later:

Kirk learns that while he and Spock were in the past, someone else was using the Guardian of Forever:

KIRK: If we didn't change anything while we were in the time vortex, someone else must have. Was the Guardian in use while we were gone?

GREY: Yes, but it was nothing unusual. We were scanning recent Vulcan history.

SPOCK: What time period?

GREY: Twenty to thirty Vulcan years past.

So the change in the timeline happened twenty to thirty Vulcan years in the past.

And in another scene:

BATES [on monitor]: Yes, sir. I can relay that to your screen. (showing relevant images) Sarek of Vulcan. Ambassador to seventeen Federation planets in the past thirty years.

SPOCK: That is not correct.

So the change in history should have been thirty or more years earlier. Unfortunately it is not specified what type of years. They could be Earth years, Federation years, Starfleet years, Vulcan years, or Time Planet years.

If the change was both twenty to thirty Vulcan years earlier, and at least thirty Vulcan years earlier, it should have been thirty Vulcan years earlier.

And:

SPOCK: My mother. The son, what was his name and age when he died?

BATES [on monitor]: Spock. Age seven.

And the records might give Spock's age in Earth years, Federation years, Starfleet years, Vulcan years, or Time Planet years.

And Spock tells the Guardian of Time":

SPOCK: I wish to visit the planet Vulcan, thirty years past, the month of Tasmeen. Location, near the city of ShirKahr.

And that could be thirty years in Earth years, Federation years, Starfleet years, Vulcan years, or Time Planet years.

If all the years in "Yeseryear" are years of the same length, Spock was probably aged about thirty seven in "Yesteryear" and probably about thirty two when he became Kirk's first officer about five years earlier.

But only if all the years were the same type of years.

Dr. McCoy's statement would probably be made in Earth years, and less likely in some hypothetical Starfleet or Federation standard year, and less likely still to be made in Time Guradian planet years, and least likely to be made in Vulcan years, since this McCoy has not be close to a Vulcan for several years.

The historians say they using the Time Guardian to study recent Vulcan history twenty to thirty Vulcan years ago. So that time span is in Vulcan years.

Spock asks the Time Guardian to send him to Vulcan thirty years in the past, without specifying which type of years, which seems like a big oversight.

If those are thirty Vulcan years, then Spock should go to a time between 30.0 (thirty point zero) and 31.0 (thirty one point zero) Vulcan years in the past.

But if those years are not Vulcan years but Earth, Federation, Starfleet, or Time Planet years, then the relative lengths of Vulcan years and those years can be calculated.

If 20.0 (twenty point zero) to 31.0 (thirty one point zero) Vulcan years are equal to 30.0 (thirty point zero) to 31.0 (thirty one point zero) other years, a Vulcan year must equal 0.967 (zero point nine six seven) to 1.550 (one point five five zero) of those other years.

And it is natural for Star Trek fans to assume that when unspecifed types of years are used, they are usually or always Earth years, and thus that the unspecified other years in this episode were Earth. years.

If the thirty years in the past in "Yesteryear" were Vuclan years, Spock would have an age of about 37 in Vulcan years in that episode.

"Yesteryear" may be a few years after "Journey to Babel" and "The Enteprrise Incident", and Spock has apparently been a Starfleet officer for 18 (eighteen) years in those two episodes. Those unspecified years would usually be assumed to be Earth years.

Assuming that Spock was at least 18 (eighteen) Earth years old when he became a Starfleet officer, Spock should have been aged at least about 36 (thirty six) Earth years in "Journey to Babel" and in "The Enterprise Incident".

If Spock was at least 18 (eighteen )when he was commissioned an officer in Starfleet, he would have been at least 31 (thirty one) in "The Menagerie", and probably years older, since it would probably take Spock at least a few years to become third in command of the Enterprise, as he seems to have been at Talos IV. If Spock was aged 22 (twenty two), the usual age for graduating from US service Academs in the last one hundred and fifty years, he would be aged 35 (thirty five) in "Menagerie" plus the number of years it took Spock to become third in command of the Enteprise.

On the other hand, spock's human mother Amanda doesn't look very old in "Journey to Babel". Jane Wyatt was born August 12, 1910, and was fifty seven years, one month, and thirteen to sixteen days old when her scenes were filmed. My copy of the script and The Making of Star Trek describe Amanda was being 58. If I remember correctly, Amanda was described as thrity years old when Spock was seven in "Yesteryear".

If Amanda was sixty years old when Spock had been a Starfleet officer for eighteen years, she would have been forty two when Spock became a Starfleet officr. If Spock was eighteen when he became a Starfleet officer, he would have been born when Amanda was about twenty four and would be thirty six. If Spock was twenty two when he became a Starfleet officer, he would have been born when Amanda was about twenty and he would be forty. If Amanda was only fifty eight when Spock had been in Starfleet for eighteen years aged thirty six to forty, she would have been aged eighteen to twenty two when Spock was born.

And nobody knows how long it may have taken for Spock to be born after Amanda married Sarek. So the Sarek-Amanda relationship is beginning to have some of the creepiness of Edward and Bella in the twilight series, or the historical marriage between Emperor Andronikos I and Agnes/Anna of France.

So any Star Trek chronologist is going to have to choose whether the TOS era equivalents of tabloid newspapers had headlines like "Alien Ambassador Elopes With School Teacher One Third His Age" or "Precocious Half Human Child Enters Starfleet Academy at Half the Usual Age", or maybe give part of the precocity to Amanda and part to Spock.

Without assignng great precocity to Amanda and/or Spock, it would be hard to make Spock much older or younger than his mid to late thirties in TOS. And spock's age in "Yesteryear" seems to be about thirty seven. So if Spock was thirty seven Vulcan years old in "Yesteryear", the length of a Vulcan year should be very close to an Earth year.

If Spock went back in time 30.0 (thirty point zero) to 31.0 (thirty one point zero) Vulcan years to when he was aged 7.0 (seven point zero) to 8.0 (eight point zero) Vulcan years, His age in "Yesteryear" would be between 37.0 (thirty seven point zero) and 39.0 (thirty nine point zero) Vulcan years.

If Spock was about 36.0 (thirty six point zero) to 41.0 (forty one point zero) Earth years old in "journey to Babel", and if "yesteryear" should have been less than 6.0 (six point zero) Earth years later, Spock's age in "yesteryear" should be between 36.0 (thirty six point zero) and 47.0 (forty seven point zero) Earth Years.

And a Vulcan year would be between 0.923 (zero point nine two three) and 1.270 (one point two seven zero) Earth years long.

What if Spock went back in time 30.0 (thirty point zero) to 31.0 (thirty one point) Earth years to when he was severn years old.

If 20.0 (twenty point zero) to 31.0 (thirty one point zero) Vulcan years are equal to 30.0 (thirty point zero) to 31.0 (thirty one point zero) Earth years, a Vulcan year must equal 0.967 (zero point nine six seven) to 1.550 (one point five five zero) Earth years.