Celestial Nav?

[StuM]

Originally Posted by Mike1956

Have just found an online calculator and did the calculation (twice) and my answer is different to 8 billion miles. As I intimated these things are generally beyond me.

Calculation as follows

5878625373183.6 x 8.611 =

x 2 =

x 22 =

divide by 7 =

divide by 360 =

divide by 60 =

14730933173.9 MILES

Any idea what I have done wrong?

Thanks

Mike

What you've done wrong is assumed that precision and accuracy are the same thing.

You have the number of miles in a light year to 14 digits, but then use " x 8.611" ( 4 digits) as an approximation for the number of light years to Sirius That means that your distance in miles is only accurate to 4 digits at best.

You then use " 22 / 7 " as an approximation for pi. That further reduces the accuracy of your result.

Consequently your very precise "14730933173.9" is actually only accurate to one or two digits. Anything beyond that should be rounded to zeros.So the best you can tell from your calculations is that the answer is "somewhere around 15000000000 +/- 500000000"

Which is what Kelkara and I also came up with

[Cavalier]

355/113 is a much better approximation for Pi (if one must be used) than 22/7. The former is good to 6 decimals places, whereas the latter is only good to 2.

Sent from mTalk

[StuM]

Originally Posted by Cavalier

355/113 is a much better approximation for Pi (if one must be used) than 22/7. The former is good to 6 decimals places, whereas the latter is only good to 2.

Sent from mTalk

And 3.141592653589793238462643383279502884197169399375 1058209749445923078164062862 is an even better approximation

But neither of them can improve the accuracy beyond the precision of the 8.611

[Mike1956]

I feel I have now managed to determine as accurately as I am able the solution to my problem. (to be fair someone else has done it for me)

I said in my original post "I am no mathematician" this is true.

I also said during the thread I am unlikely to be able to use an online calculator. This indeed has proved to be correct as STUM has pointed out. Anyone who knows me considers me to be the village idiot which I am entirely comfortable with and is I guess pretty accurate.

However to the problem

When I attempted to use the online calculator I arrived at a figure of 14,730,933,173.9 miles (of the 1760 variety not the 1852)

Other persons answers were given in distances to other planets which required additional calculations to determine miles. Apparently they calculated to be around 8 billion miles. We now have a discrepancy of around 7 billion miles.

As navigators whether there is doubt or not we are always seeking additional information to confirm our position. In this case there is considerable doubt so.....

I rounded the numbers involved and did the calculation by mental arithmetic as follows

6 trillion x 8 = about 50 trillion

x2 = 100 trillion

x3 (forPi) 300 trillion

divide by 360 = about 900 billion

divide by 60 = 15 billion

This is a rough calculation which seems to confirm my figure and casts doubt on the 8 billion.

I then enlisted a friend and put the problem to her.

Mentioning Sirius, Earth and our Sun is of course irrelevant.

The problem is we have a circle with radius 50.620,843,088,484 miles with an angle at the centre (a) of 1 minute with the radiuses extending to the circumference meeting the circumference at (b) and (c). We need to know the distances from b to c along the arc and in a straight line.

My friends calculations indicate

14,725,006,400 miles

and

14,725,006,370 miles

This confirms my problems using the online calculator as I am 5 or 6 million miles out. Not sure if my crude approximation for Pi might be the culprit.

It also casts further doubt on the figure of about 8 billion miles suggested earlier in this thread which is around 7 billion miles out.

Mike

[StuM]

Don't know where your figure of 8 billion comes from. We all agree that it is about 15 billion miles.

Any more precision than that is fairly meaningless since no one knows exactly how far away Sirius is give or take 5 billion miles.

At best we know that it is on average somewhere around 50.62 trillion +/- 5 billion miles away. And to add to the problem Sirius is actually a binary star which is itself in orbit with another small dense star about a central point - so it's distance from us also varies by some fraction of it's orbital radius depending on when you try to measure it - that could be another 1 -2 billion miles variation.

Plus, depending on where the Earth is in it's orbit, the distance can also vary by around another 180 million miles.

Your apparent 5 or 6 million "out" is totally meaningless given the imprecision of the distances involved.

[Kelkara]

OK, I'll give this one more try ...

5878625373183.6 x 8.611 x 22 divide by 7 divide by 360 divide by 60

This is the sum you first asked about, and the answer is 7,365,466,585.9487271904761904761905 to as many decimal places as my windows 7 calculator will give. This is the only number that is close to 8 billion that anyone has calculated on this thread ... but you quickly pointed out in your next post that you forgot to multiply by 2. WHich is why until now nobody has given you the answer to this calculation.

If you multiply by 2 this is the new answer:
14,730,933,173.897454380952380952381

which differs from your calculation of 14730933173.9 by 13ft 5¼"

Other persons answers were given in distances to other planets which required additional calculations to determine miles. Apparently they calculated to be around 8 billion miles. We now have a discrepancy of around 7 billion miles.

I read through this entire thread again, and nobody anywhere has told you that the answer is about 8 billion miles. Forget the number 8 billion. You made it up. It doesn't belong here. You now have Me, StuM, your friend and yourself all saying the answer is close to 15 billion miles. How many people do you need to do this calculation for you before you believe the answer?

I feel I have now managed to determine as accurately as I am able the solution to my problem.

No, as StuM has pointed out, what we have now done is perform the calcuation as PRECISELY as possible, not as ACCURATELY as possible.

Accuracy concerns errors, so let's talk about the errors in the calculation.

The first error is that the calculation itself is an approximation - you are calculating the distance around a curved arc, whereas the shortest distance would be a straight line ... so how significant is this error?

The shortest straight line distance is 14,725,006,159 (to the nearest mile) which is 5,927,015 miles shorter.

But let's just look at the errors in your calculation...

first: you used 22/7 as an approximation for pi. 22/7=3.1428571... but pi is really 3.1415926535... SO how much difference does this make? The same calculation, but using the correct value for pi gives 14,725,006,367, which is 5,926,807 miles shorter than your calculation.

But it is only 208 miles longer than the straight line calculation, which means that the approximation for pi is about 30000 times more significant a source of error than performing the calculation along a curved arc not a straight line.

The next error is the accuracy of the distance to Sirius 8.611 lightyears. This means somewhere between 8.6105 and 8.6115 lightyears. so what effect does this have on our answer?

This gives a range of answers from 14,724,151,355 to 14,725,861,378. An uncertainty of 1,710,023 miles - which is three times as significant an error as the error introduced by your approximation of pi.

This is the real answer to your question ... 14,725,000,000 +/- 850,000 miles.
But that is just using the numbers ... As StuM said we don't actually know where either earth or sirius really are.

But ...

Since this is a question about Celestial Nav, the final error is the accuracy that you can read 1 minute of arc on your sextant. If your sextant is anything like mine, then I can get a little bit better than 0.5 minute accuracy on the vernier (not counting other sources of error like temperature). So 1 minute of arc as measured on the sextant is actually somewhere from 0.75 to 1.25 minutes. How important is this error on the calculation?

This will give a range on answers from 11,043,754,775 to 18,406,257,958. This is an error of 7,362,503,183 miles, which is 4000 times more significant that the uncertainty in the distance to Sirius.

I don't think that I can answer your question any more completely than that, so this is my last word on the subject.

The answer isn't exactly 15 billion, it is closer to 14.7 billion (If you have some kind of super-sextant), but unless you're trying to teach the voyager spacecraft to navigate using a sextant, 15 billion is close enough.

I hope it has helped somewhere.

[Mike1956]

Thank you all,

I must apologise for causing you all this additional explanation. I too have just re read the thread and there is no mention of 8 billion miles anywhere. I reckon all these long numbers have got to me. Seems like my original calculation using the dodgy figure for Pi and my ineptitude using the online calculator was close enough after all. It is certainly close enough to satisfy my curiosity.

Once again many apologies

Mike

[Kelkara]

Originally Posted by Mike1956

Thank you all,

I must apologise for causing you all this additional explanation. I too have just re read the thread and there is no mention of 8 billion miles anywhere. I reckon all these long numbers have got to me. Seems like my original calculation using the dodgy figure for Pi and my ineptitude using the online calculator was close enough after all. It is certainly close enough to satisfy my curiosity.

Once again many apologies

Mike

No Problem mike,

but what I hope that I have showed is that you displayed no ineptitude using the online calculator (you got the right answer), and the value of pi was a perfectly good approximation compared to the other errors (so for most things you can keep on using 22/7 for pi if it helps you).

All you need is a little more confidence in your own ability to get it right - a true "village idiot" won't even start trying to do navigation calculations.

[StuM]

Originally Posted by Kelkara

No Problem mike,

but what I hope that I have showed is that you displayed no ineptitude using the online calculator (you got the right answer), and the value of pi was a perfectly good approximation compared to the other errors (so for most things you can keep on using 22/7 for pi if it helps you).

All you need is a little more confidence in your own ability to get it right - a true "village idiot" won't even start trying to do navigation calculations.

Just to add,

Mike - I hope you have also learnt a little about spurious precision, error ranges and how much to trust quoted numbers.

It's useful knowledge which helps in interpreting the validity behind election polls and global warming claims.

[Mike1956]

Originally Posted by Kelkara

You've done nothing wrong. That is the correct answer ... 15 billion miles.

now you're starting to make me doubt myself.

Thanks StuM and Kelkara for your kind words of encouragement.

Have just had another look through the thread looking for the "missing 8 billion miles post" and found this post. I am wondering why the doubt?

Mike

[Kelkara]

Originally Posted by Mike1956

Thanks StuM and Kelkara for your kind words of encouragement.

Have just had another look through the thread looking for the "missing 8 billion miles post" and found this post. I am wondering why the doubt?

Mike

The doubt was mostly metaphorical ... After you were explicitly told the answer twice by two different people using two different methods and confirming your method also gave the same result - you then questioned not once but twice the veracity of the answer.

but I too am human and able to make mistakes ... The more times you tell me I'm wrong the more likely I am to question whether the mistake is indeed mine ... a classic technique for getting confessions out of innocent people.

[Mike1956]

Just to reiterate....

This thread is about completing a calculation to confirm that rays of light from stars can be considered to be parallel for navigational purposes when they arrive at Earth. Chose Sirius as it is comparatively close and managed to determine 1 minute of arc at Earth is about 14.75 trillion miles.

Just realised we did not complete the final part of the calculation. Clearly the rays of light are not parallel when they arrive so what maximum inaccuracy are we ignoring?

The calculation required is something like.....

Diameter of Earth orbit around the Sun x 1760 (assume light years are statute miles) Divide by 14,725,006,400

I reckon the answer is about 20 metres?

Stars further away from Earth than Sirius will have smaller maximum errors.

Mike