Formula for calculating aging rate of a decaying half-elf dynasty

Question

A powerful human king had a child with an elf princess. His life expectancy is about 70 years (human average), hers is 2000 years. The half-elf prince is expected to live for about 1000 years, more or less the average of his parents life expectancy.

Years later, the prince has two children: a boy, with an elven lady; and a girl, with a mortal woman. Both of them, having a half-elf father, are gifted with a long live, but the boy lives longer than his sister, since he's the son of an elf and she's the daughter of a mere mortal.

The prince becomes king, and his son becomes king after him, and so on. Their descendants have children with elves and also with humans (increasing and shortening their lifespans respectively). After generations of mixing with humans, the powerful half-elven kings lose the grace of the magical folk and become simple mortals, with an average life expectancy.

What I've been doing until now is simply calculate the average and give them linear aging. But this yields certain problems... The prince's son, for example, would have a life expectancy of (1035+2000)/2 = 1517.5 years. With linear aging (0 looks 0, 1517,5 looks 70-80), this poor boy would be over 300 years old before even reaching puberty. If the same happens with pregnancy, he would be in her mother's womb for over 15 years!

So, I want to know if there's a mathematical formula I can use to calculate their aging rate taking into account that they should reach puberty relatively early on their lives and lengthen or shorten their life expectancy depending on whether their mothers are elves or mortals.

PD: Of course all of this is fantasy and I can give them the life expectancy I want, but I'd like to keep it more or less realistic. Also, having a "consistent framework" to work within makes me feel more confortable creating. Thank you very much!

Answer 1

Exponential

Mathematical formula? You got it! I'm going to call this the simplest nonlinear mapping, unless someone would like to challenge me on that claim.

If you have a desired life expectancy, you can map the human lifespan onto it with a power relation, of the form

$\text{(human-equivalent age)}^{P} = \text{elf age}$

$P = \log_{10}{\text{(elf life expectancy)}} / \log_{10}{\text{(human life expectancy)}} \approx \frac{\log_{10}{\text{elf life expectancy}}}{1.85}$

^{(ok fiiine you can use whatever log base you like)}

Neat. What's this mean?

It looks to me like you already have a means of calculating the life expectancy of someone with elf blood — average that of their two parents. I don't really see any major problems with this. "My mom's life expectancy is 1000 and my dad's is 2000, so mine is 1500" does not obviously sound wrong. I'll let someone else come up with a cleverer approach, but there's not much real-world basis for elf lifespans, so we're all making stuff up here.

Take the log-base-10 of that, and divide by 1.85 (the log-base-10 of about 71, which is a lowballed human life expectancy, but you can adjust). Many calculators have a button for log-base-10, or you can type "log10(1500)" into WolframAlpha and get just around 3.18. For our 1500-year-life-expectancy mortal, his exponent P is log10(1500)/1.85 $\approx$ 1.717.

How old does an elf-blooded person need to be to look age X? They need to be X to the power of P.

• Our prince's son is probably out of diapers by something like 2.5^1.717 = 4.8 years old.
• He enters human-like adulthood at 18^1.717 = 143 years of age.
• He might begin his physical decline soon enough, at 30^1.717 = 344, but he's got a long way to go.
• His expected lifespan does turn out right, at 71^1.717 = 1509 years (alright, there were some rounding errors, close enough).
• If he's lucky and makes it to the elf-blooded equivalent of a human centenarian, he'll have lived an impressive 100^1.717 = 2716 years.

One minor problem: the power law would imply that elves grow faster than humans before age 1, since, say, our guy looks six months old at 0.5^0.717 = 0.3 years or about four months of age. You have some options here: roll with it, or just lock the first year to human aging and then apply the power law after age 1.

Lot of words for a simple concept, but I'm happy to explain further.

Answer 2

I'm going to give the cheat answer:

There's no reason why extraordinary long life has to include an extraordinary long gestation and maturation period.

Now, for a bit of science-y stuff for why I'm going to give my cheat answer: We sort of know that when you are young, certain gene pairs turn on and turn off as part of the transition from child into adulthood.

We also know that certain gene-pairs turn off as we age.

It's therefore entirely possible that the Elven genome has pairs that essentially keep their body in a prime state by not turning off those gene-pairs that contribute to ageing in the same way that humans do, whilst not affecting the development from Infant to adolescent to mature adult.

If we go with a evolutionary explanation: Elf children still need to go from Baby to Child quickly enough in ye olden times to avoid being preyed upon by predators.

If we go with the humorous evolutionary explanation, As a Parent - nothing would make me want to throw my kids out of the window more than I already do, than them staying that way for a decade. :D