Okay, I’m not quite that bad. Yes, I was was a Music Major, but I really loved Music theory, which is mostly math using a different set of symbols and rules. Likewise, I loved my accounting classes, and the math-y parts of chemistry. Really, anytime Math was connected to something non-abstract, I did well with it. But when Math was all abstract numbers and fiddling with abstract numbers. . . um, not so good.

So often as students, we ask our teacher: When are we gonna use this in real life? The answers were generally so terrible I can’t even remember what they were.

I, however, have encountered a sudden need to use Trigonometry, something I haven’t even *seen* since, like, *high school*.

My problem is as follows: I’m writing a science fiction novel. Well, it might be argued that it’s more fantasy set in the future, with spaceships and such. . . like ninjas. Anyway, the elder brother (Thomas) of one of the main characters (his sister Niut) needs to get home to his sister ASAP. He is currently on the space station Birkeland 3, orbiting the star Caph (aka, Beta Casseopiae). He needs to get to the planet Fatima, orbiting the planet Hamal (aka, Alpha Arietis), where his sister Niut is. In the scene I am writing, he is meeting with the local “Station Chunin” (A chunin is literally a “middle man”, in this case, a local manager/ organizer sort of person) to plan his way home. An important part of the discussion is mention of the distance between these two stars– I have found one calculator, but I want to be sure I understand the math behind it, so I can try it myself if needed later (the site goes down, I loose the bookmark, etc.)

I get the equations from here: http://www.neoprogrammics.com/distance_between_two_stars/

I get the basic data for the stars from their respective Wikipedia entries (which is pretty decent for this sort of work).

Converting coordinate hours, minutes, seconds to decimals occurred here: http://www.robertmartinayers.org/tools/coordinates.html

Let’s see me attempt this on my own. . .

It looks like the basic idea is to take the known coordinates, translate them into X,Y,Z coordinates, and then plug those into a final equation. We need the Right Ascension (a), Declination (d), and Distance in Light years (R).

X= R * C0s(a) * Cos(d)

Y= R * Sin(a) * Cos (d)

Z= R* Sin(d)

With Star 1 labeling X1, Y1, Z1, and Star 2 labeling X2, Y2, Z2.

Caph (Star 1)

**X1**=54.7 * Cos(2.2945) * Cos(59.1498) =54.7 *.999198243*.51279525 **= 28.0274**

**Y1**=54.7 * Sin(2.2945) * Cos(59.1498) = 54.7 * .040035877*.51279525**= 1.1230**

**Z1**=54.7* Sin(59.1498) = 54.7 * .858510938 **= 46.9605**

Hamal (Star 2)

**X2**= 65.8 *Cos (31.7934) *Cos (23.4624)=65.8*.849953388*.917321554**=51.3029**

**Y2**=65.8*Sin(31.7934)*Cos(23.4624)=65.8*.526857891*.917321554**=31.8010**

**Z2**=65.8*Sin(23.4624)=65.8*0398147168**=26.1980**

Now, we find the differences of each coordinate:

**dX**=X2-X1= 51.3029-28.0274**=23.2755**

**dY**=Y2-Y1=31.8010-1.1230**=30.6780**

**dZ**=Z2-Z1=26.1980-46.9605**=-20.7625**

We plug these three numbers into the following equation to find for D (distance in light years):

D=SqRoot (dx*dx + dy*dy + dz*dz)

D=SqRoot (23.2755*23.2755 + 30.6780*30.6780 + -20.7625*-20.7625)

D=SqRoot (541.7489 + 941.1397 + 431.0814)

D=SqRoot (1913.97)

**D=43.7489**

**Therefore, the distance between Beta Casseopiae (Caph) and Alpha Arietis (Hamal) is 43.7489 Lightyears.**

Checking against the calculator, this looks right.

Grunt (and other truly mathy sorts reading this): if you were my teacher, would you think I understood this process, based on the work shown?

Nice work, Zoph! You’re not really Math-Hating-Barbie after all. This looks really good on the surface. I need to check the math later after I get off work, but it looks right. The only thing I can think of to ask right off the bat, and this might be explained in your Wiki link, is whether you’ve accounted for motion of the stars between the current time and the future time your story occurs in. If they tell you how fast (and the direction) they’re moving, you can calculate what the distance will be at the time of your book. Maybe they take that into account in the source though. I haven’t checked yet.

Nope, didn’t take that into account. . .I can try to work that out in a bit. . .(it’s dinner time, y’all, and I need eggrolls. . .)

And saki. π It’s good for the brain.

lol. I dunno. I may go for the whiskey for this . . .

ok, Grunt. . . I’m stymied on this–this is where my math weakness really comes out. I’m finding velocity numbers, but I’m not sure which ones to use.

Also, not entirely sure how to use them. Would I take the number (say, N=radial velocity, and you scale N up to units of a year, and then multiply my number of years in the future (f). . .and then divide by 9.5 trillion (if we’re in metric) to get the delta for R above?

And then with the new R for each star, re-work the equation? Or would I need to also calculate new (a) and (d)? . . . using proper motion (m)? maybe?

So, if we’re looking at B Cas roughly 700 yrs in the future:

Radial Velocity (Rv) = 11.3 km/s, so that scales up to 356,356,800 km/yr. at 700 yrs, that’s 249,449,760,000. (I note immediately that this is significantly less than 1 light year). Divided by the km in a light year, 9.5 tril, that makes a difference of 0.0263 increase in R for B Cas.

And then RA changes via Proper motion (m) by 700yrs*523.5mas=366,450mas=366.45arcsec, or 6.1075′ (arcminutes). So the RA in 700 years would be changed

from 00 09 10.68518

to 00 15 11.76 ? (grimacing. . .)

and do the same with declination? using the (m) for dec?

I’m really guessing here. . .

I get it now. Good job on the radial. So it’s a small amount. That’s good. I think you’re right on the RA & Decl as well, but the diff looks too big.

Can I catch up with you later after I deliver a software fix at work? Maybe now’s a good time for you to try that whiskey? π

We’ll catch up tomorrow. I’ve been up waaaay to late the past 2 nights, so I gotta head to bed now if I want to be able to use English or basic addition tomorrow. . .

It was a late night, but got all my stuff done. So, as soon as I get up and clear any phone calls, I’ll get back on this. Sorry I don’t have anything new for you this morning. But I will later Wed.

Sorry about being slow to respond, but I’m still at work.

The velocity correction is more physics than math, and we can work through that, but first we need to be sure of what we have. Do you only have radial speeds away from Earth? Or do you also have up-down, sideways components? I’m guessing most of the velocity is away from us, but they might also have some smaller cross-range components.

But just looking at your radial component, yes, basically you use:

Delta-R(radial) = Velocity(radial) * Delta-Time-into-future

Which is exactly what you’re describing. Now, I think you’re suggesting units like this:

DR(light-years) = N(meters/year) * DT(years) / c(xx meters/yr)?

I don’t get the 9.5 trillion. What is that? Wait, I see you’ve posted something else. Let me reset…

9.5 trillion km to the light year. . .

Also: take your time. I gotta get to bed. I’ll check what you write in the morning. . .

Thanks for helping me with this, Grunt. Considering how fuzzy some of the rest of what I’m writing will be (shino-Catholic metaphysics! electric universe! alchemic references!), I want things like this to be all the more solid and check-able . . .

No worries! G’nite! π

Zoph, I’m just getting a few minutes to look at this again. I cranked through your vector diff between the positions of Caph and Hamal, and I confirm your answer of 43.75 LY roughly.

Working the velocities…

Here’s something right away. You’re using the number of seconds per year based on 365 days/yr, right? I know this is nit-picking, but most of the time, the number used is 365.25 days/yr, since over time, that’s a pretty accurate value for the sidereal year. The 0.25 takes into account the leap day every four years that brings the calendar into line with the actual sidereal motion of the earth around the sun. So, I would go with 31,557,600.0 sec/yr just to be “standard,” but obviously it’s not a big effect. It still gives me DeltaR=0.0264, and a new R=54.726 LY.

Ok, I confirm how you were going about calculating the deltas in RA and Declination, and I was wrong about your new RA for Beta Cas looking too big. It was just about right. Only thing is, I think you made a slight conversion error when you went from arcsecs to minutes. I get 15 min, 17.135sec, where the minutes went to 15 = 9+6, but the seconds went to 17.135 = 10.685 + 6.45. The 6.45 is the remainder of the minutes calc converted back into seconds.

But basically, all the deltas can be calculated this way, except that the declinations are expressed in angles rather than the clock-based hr-min-sec that RA is measured in. That’s just a customary thing. Otherwise it’s the same.

365.25? D’oh! Nah, I just went straight to 365. Maybe not a big effect in this example, but over a greater period of time, or with a different pair of stars, could be more significant.

I’m not suprised I made a conversion error– byt the time I was working that out, late-night-fatigue was causing cranial shut-down, and that last bit was more mental darts thrown at a wall . . .

Ok, here’s what I would do. Even though it’s a straightforward thing to calculate out the deltas like we’ve started to do, I don’t think that will give us a meaningful answer, since the radial distance to both stars has a high uncertainty of +/-0.3 LY. That’s one of the things about stars. We can measure the position in the sky (RA & Decl) to very high precision, but the distance is only estimated using some vague assumptions, and there’s even some question about those assumptions.

So, if we do our delta calcs, we’ll end up adding our 0.05-0.1 LY to the positions of each, and it will increase the distance between the stars, since if you look at the Wiki entries and figure it out, it turns out they’re both moving away from each other, so we’ll change D=43.75 to D=43.89.

But that is a little pointless, because we know the radial position of each is uncertain to 0.3 LY. The actual distance now or in 700 years could be 43.5 LY, or it could really be 44.1 LY. So, what I’d do, is take your nominally calculated value of 43.75 LY and round up by a number that’s on the same order as the radial uncertainty, since we know that the 700yr motion is less than that.

For example, I would round the distance up to 43.9 LY. That adds a substantial 0.15 LY to the original calculation, which splits the difference between the 700yr motion of probably about 0.05-0.1 LY and the uncertainty of +/-0.3 LY.

Does this make any sense?

yeah, actually, it does make sense. . .

And I’m printing out this whooooole post + comments to put into my notebook, so I can remember this later if needed.

Thank you so much for your help on this, Grunt!

De nada, Z. That was fun! π

And incidentally, for a music major, you did some pretty serious math here. Not only did you approach it very logically, but you communicated it with a clarity that made it easy for me to follow. Better than a lot of rocket scientists I know. I guess that sci-fi fangirl stuff pays off, huh?

Well, the nerdy girl, plus having a lawyer as a father coupled with the catholic-high-school experience. . . actually, this sort of hits on something I’ve been thinking about since I first posted this, and why is it, exectly, that I consider myself bad in math? It has a lot to do with how the US school system (even private schools) teach math and science, and maybe why we’ve been losing our hold on those subjects. . .

No kidding. Could it be that when the kids approach it logically, they slap your hand and tell you to forget logic? “Just follow this psycho recipe that doesn’t make sense and do as you’re told.” You might be onto something there.