L[ight] y[ear], P[arse]c, and E.T.

Inspired by this thread:

Roundabout the orbit of Saturn does a lightyear equal a parsec.

Does that change the orbital distance if the star is a different mass? Smaller or bigger?

Would it make sense to calculate or search for astronomic data using the Ly:Pc distance as a Universal Yardstick…something extraterrestrial sentience would recognize, the way we discovered and use the Planck Scale?

How different would the math be if, instead of using 360deg:60arcmin:60arcsec, we used a decimal, octal, or heximal degree system?

Please try to use informative thread titles. I have edited the title to indicate the subject.

General Questions Moderator

Saturnians probably use micro radians rather than 1/3600ths of a degree as their primary measure of angular separation.

Of course the angle used, 1 second, is completely arbitrary and obscure here. So’s a microradian.

If you want to choose a tiny fraction, or one over a big number, what dimensionless big number should you choose that other civilizations might also choose?

It won’t be ten raised to an integer power, because nothing is special about ten (unless you evolved with 10 fingers).

Maybe 2^(2^(2^2)) or 65,536 is a good choice, because 2^(2^2) is too small and 2^(2^(2^(2^2))) is much larger than necessary.

A light year is roughly six trillion miles; a parsec, about 3.26 light years. They are astronomical measurements based on Earth.

The distance light travels in the time needed for one solar orbit and the distance equivalent to a parallax of one arc second are equal when applied to a - probably imaginary - asteroid orbiting at a distance about 104% of Saturn’s orbit. They are not a “light year” or a “parsec” except in the very loose sense used in the other thread asking if there were ever a case where those two modes of measuring were equal.

(That’s not a snark at Enola Straight or his question, but making the point that the terms have specific technical meanings in English, derived from Earth-based measurements but independent of them. The cores of the Milky Way and Andromeda galaxies, for example, are roughly two million light years apart, with absolutely no reference to Earth.

I figure, if the ancient Babylonians and Egyptians found a number ( which just happens to roughly equal the number of earth days in an earth year) and was divisible by 2,3,4,5,6,…

Also, dosen’t deg:min:sec arise naturally and automatically when computing formulae in trig and analytic geometry…specifically where
Ax^2 + Bxy + Cy^2 +Dx +Ey +F=0 where B /= 0

You figure, if they found a number, … what? Are you concluding it’s a universably favorable number? Because of course the number of days in a year is also completely Earth-centric thinking.

The “divisible by” argument may favor numbers whose factors are the first few primes, such as 210 or 2310. There aren’t any of these between 210 and 2310, though. For example, being divisible by 6 is no achievement if we already specified divisibility by 2 and 3.

Where does deg:min:sec arise naturally? Your equation wouldn’t make dd:mm:ss pop out, except if you had already snuck them in with earlier conversions from radians.

Aha, okay.

Well anyway…

would the Saturn-oid Ly:Pc orbit remain the same regardless of the mass:size of the star?

No. The orbital period is a function of the distance between the two bodies and the sum of their masses. In practice, stars generally have so much more mass than planets that the latter mass can usually be ignored. Two planets at a given distance from two different suns will have different local years, with the planet of the more massive star having a shorter period. That will change the length of their respective light years.

The formula is

P = 2pi * sqrt(a^3/G(M + m))

P = period
a = semi-major axis (distance between the star and planet)
G = gravitation constant
M = mass of star
m = mass of planet

We can establish a formula that describes the relationship between a and M+m in a system that has this coincidence where 1 light-annum = 1 parallax distance measurement.

Let’s continue from the above and define a few more terms. I think I did all the algebra right; could someone please check me?

c = speed of light
s = angular measures in a full circle (Babylonian arcseconds = 3606060 = 129600)
S = Parallax distance measurement, AKA “local parsec”

S = a * π * s

L = P * c ( 1 “light annum” is the distance light travels during 1 orbital period.)

L = S ( This is our coincidence we’re solving for.)

I’m gonna throw out the m term for the time being, since it’s usually negligible. Since M and m are always considered together, we can put m back in at the end if we want.

Now, solve for a and M.

a * π * s = 2 * π * sqrt(a[sup]3[/sup]/(GM)) * c

a * s = 2 * sqrt(a[sup]3[/sup]/(GM)) * c

π falls out right away. :slight_smile:

s = 2c * sqrt( a / (GM) )

s[sup]2[/sup] = 4c[sup]2[/sup] * a / (GM)

a = GM * s[sup]2[/sup] / (4c[sup]2[/sup])

M = a * 4c[sup]2[/sup] / (G * s[sup]2[/sup])

Did I do that right?
Now, here’s the real question: Assuming the star is a main-sequence dwarf, what’s the range in values for M that result in the planet orbiting within the life zone? :slight_smile:

*** Ponder