Mathematical genius required

[Mike53]

Hi all,

Just got in and catching up with this.

Firstly I must apologise. My distance to Sirius was from memory and it is incorrect. I said 8.11 lyrs and have just googled "distance to Sirius" and the first result was 8.611 lyrs. SORRY!. I certainly accept and understand the distance to Sirius will vary by +/- 180 million miles due to Earth rotation but at the speed of light this is only about 15 minutes and for this calculation can be safely overlooked.

Now I have my apology out of the way I will try and get my head round the maths.

Thanks

Mike

 

[EFLImpudence]

Just out of interest, why have you chosen Sirius?

It is the brightest star but the nearest star is about half the distance.

 

[Mike53]

Just out of interest, why have you chosen Sirius?

It is the brightest star but the nearest star is about half the distance.[/QUO

This is all about celestial navigation. I read about the rays of light from stars being parallel which they clearly are not and chose Sirius as it is the closest navigational star and its rays upon reaching the Earth will be less parallel than light from other stars much further away. This was a couple of years ago. I remember trying to calculate the distance of 1 minute of arc at Sirius distance, I think it was about 12 million miles. So if 1 minute of arc covered 12 million miles the Earth at diameter 12,742km was a tiny fraction of 1 minute of arc. I think I concluded that worst case scenario from one side of the Earth to the other by ignoring the divergence in the rays of light from the stars I was ignoring an error of only a few metres (7 metres I think).

Mike

 

[Mike53]

Using the new corrected Sirius distance 8.611 lt yrs I will attempt the golf ball calculation.

A lt yr is 9.461e+12 km - very unsure about this

However I guess this means 9,461,000,000,000 km

x 8.611 lt yrs

= 81,468,671,000,000 km (distance to Sirius)

divide by Earth diameter 12,742 km

= 6,393,711,426.78

Distance to Sirius is 6,393,711,426.78 times Earth diameter

SO

6,393,711,426.78 x golf ball diameter (42.67mm)

= 272,819,666,581 mm

How am I doing?

How many Km is that?

2,728,196 Km ?

Mike

 

[Mike53]

272,819 km ?

Mike

 

[SFK]

I have changed your maths order to make it simpler as I saw you first working as incorrect at the very end:


Using the new corrected Sirius distance 8.611 lt yrs I will attempt the golf ball calculation.

A lt yr is 9.461e+12 km
so 9,461,000,000,000 km x 8.611 lt yrs
= 81,468,671,000,000 km (distance to Sirius)

Scale-Factor
Calculate what scale-factor do you have to reduce Earth to make Golf Ball Sized:
= Earth diameter 12,742 km / golf ball diameter 42.67 mm
= 12,742 km / 0.042,67 m < converted to m
= 12,742 km / 0.000,042,67 km < converted to km
= 298,617,295

So to make Earth golf ball Sized you have to divide size of Earth by 298,617,295.

Likewise if Earth now golf ball sized you have to divide Earth to Sirius distance by Scale-Factor of 298,617,295:
= 81,468,671,000,000 km (distance to Sirius) / 298,617,295 (scale-factor)
= 272,819 km

SFK

 

[Mike53]

Now attempting to solve other question. What size would Sirius be if everything scaled down so Sirius Earth distance was 970 km (Lands End to John o Groats as crow flies)

New Sirius distance 81,468,671,000,000 divided by 970

= 83,988,320,618.6

Sirius distance is 83,988,320,618.6 times the distance Lands End to John o Groats

Earth diameter 12,742 km divided by 83,988,320,618.6

= 1.5171157e-7 ?

Would this be about 1.5mm ?

Mike

 

[SFK]

Mike, only wrong at very end. try this:

What diameter would Earth be if everything scaled down so Sirius Earth distance was 970 km (Lands End to John o Groats as crow flies)

Scale-Factor
= Sirius distance 81,468,671,000,000 km divided by required distance of 970 km
= 83,988,320,618.6

So Sirius distance is 83,988,320,618.6 times the distance of Lands End to John o Groats.

Earth diameter 12,742 km divided by 83,988,320,618.6
= 1.5171157e-7 km
= 1.5171157e-7 km * 1,000 <to change km to m
= 1.5171157e-4 m * 1,000 <to change m to mm
= 1.5171157e-1 mm <move decimal point to get rid of exponential
= 0.15171157 mm

= 0.151mm (or about diameter of a hair)

SFK

 

[Mike53]

Thanks SFK,

Its not difficult to see why light rays from stars can be considered to be parallel when we reduce the problem to a scale we are able to comprehend.

Mike

 

[EFLImpudence]

This is all about celestial navigation. I read about the rays of light from stars being parallel which they clearly are not and chose Sirius as it is the closest navigational star and its rays upon reaching the Earth will be less parallel than light from other stars much further away.

Unless, of course, the other stars were much bigger.

This was a couple of years ago. I remember trying to calculate the distance of 1 minute of arc at Sirius distance, I think it was about 12 million miles.

Surely that is a massive underestimate.

One arcsecond is 93 million miles at one parsec - the definition of a parsec.
1 parsec is 3.26 light years
Sirius is 2.64 parsecs from the Sun, so one arcminute will be 60 x (93,000,000 x 2.64)
Edited x2.64, not /

So if 1 minute of arc covered 12 million miles the Earth at diameter 12,742km was a tiny fraction of 1 minute of arc. I think I concluded that worst case scenario from one side of the Earth to the other by ignoring the divergence in the rays of light from the stars I was ignoring an error of only a few metres (7 metres I think).

Back to drawing board.

 

[Mike53]

Thanks EFLimpudence,

Of course you are correct. I have found my calculations from 2 years ago, working in miles then.

Sirius 50,620,843,088,484 miles (whether this is correct or not it is the figure I used.

I managed to determine 1 minute of arc was 14,730,933,173.9 miles (hope this is correct)

I also imagined I was dealing with a triangle with 2 sides 50,620,843,088,484 miles and one angle of 1 minute. My calculations reveal the side opposite to the angle of 1 minute to be 14,725,006,400 miles

I then calculated the error I was ignoring to be in the region of 20 metres, worst case scenario.

Are these figures more in line?

Mike

 

[EFLImpudence]

Yes, 50 trillion is the accepted distance.

The numbers are too cumbersome and only approximate.

Sirius is 1,250,000 miles diameter and is itself 1/168 arcsecond.

So 1.25m x 168 x 60 = 1 arcminute.

That is 12,600,000,000 so, near enough and not a million miles away - oh yes it is.

 

[EFLImpudence]

Actually my previous post 60 x (93,000,000 x 2.64) comes to 14,731,200,000 - very close to yours.

 

[ban-all-sheds]

They are actually converging - as are the Sun's.

ELFImpudance ,
Yes the light is converging from the Sun and also from Sirius

Surely light radiating from a source is diverging the whole time.

If light from a source several light years away is converging on The Earth, what is focussing it?

 

[EFLImpudence]

Do you really want me to explain?