Mathematical genius required

[EFLImpudence]

The speed of light is 300,000 km/sec.

Therefore the diameter of the Earth 300,000/12,742 is 1/23.54 light seconds.

So number of seconds in 8.11 years x 23.54 = 742,357,440

meaning the scale distance will be that number times whatever diameter you choose for your Earth.

So, 1cm. Earth means Sirius will be 7,423.57440 km away.

 

[rsgaz]

Do not understand this to the power of stuff.

In which case, let's stick with actual numbers(using the figures you gave), despite how long they are.

Sirius is 8.11 lt yrs away

9460730472580800*8.11=76726524132630288 metres

1. What would be the size of the Earth if the distance to Sirius was 970km

So, the scale you are using is 76726524132630288/970000=79099509415 (how many decimal places do you want?! There's lots!)

Divide the earth's size by that ratio...

12742000/79099509415=0.000161088230 metres, which is 161.088 microns.

 

[Ian H]

For Q1,

81,460,000,000,000 / 970 = 83,979,381,443.

12,472 x 1,000,000 = 12,472,000,000mm

12,472,000,000 / 83,979,381,443 = 0.1485mm

So my answer to Q1 is 0.1485mm

****** IGNORE THIS, I GOT THE EARTHS SIZE WRONG******

 

[SFK]

Regarding exponetials.......
When I say "2.4E6" teh number after the E is basically the number of zeros you have to add or remove from the number.

If the number after the E is positive you multiply the number before the E by a 1 followed a number of 'zeros' equal to the number after the E.
So 2.4E6 is 2.4 x 1,000,000 (six zeros so 1million), so you get 2,400,000.

If the number after the E is negative you divide the number before the E by a 1 followed a number of 'zeros' equal to the number after the E.
So 2.4E-3 is 2.4 / 1,000 (three zeros so 1thousand), so you get 0.0024

It makes calculators and excel much easier with big numbers as on a calculator for 2.4E6 you simply type: 2.4 Exp_Button 6.
And for for 2.4E-3 you can simply type: 2.4 Exp_Button +/- button 3



My calcs without Exponentials.

1. What would be the size of the Earth if the distance to Sirius was 970km?
New Dia of Earth = Dia of Earth * (New Distance to Sirius / Distance to Sirius)
New Dia of Earth = 12,742km * (970km / 76,700,000,000,000 km)
New Dia of Earth = 0.000000161 km
New Dia of Earth = 0.000161 m
New Dia of Earth = 161 μm

The equation if using Excel would be (you can copy and paste this into excell) "=12742*(970/7.67E13)"


2. What would be the distance to Sirius if the Earth was the size of a golf ball?
New Distance to Sirius = Distance to Sirius * (New Dia of Earth / Dia of Earth)
New Distance to Sirius = 76,700,000,000,000km * (0.00004267km / 12,742km)
New Distance to Sirius = 256,850km

The equation if using Excel would be (you can copy and paste this into excell) "=7.67E13*(42.67E-6/12742)"

 

[SFK]

Mike,
Ian says answer to Q1 (New Dia of Earth) is 0.1485mm and Rsgaz and I said 0.161mm, so close enough to be the same as we are using slightly different distances for distance of Sirius.
SFK

 

[Ian H]

Off the proper distance (not my google one). I get:

Q1:
76,700,000,000,000 / 970 = 79,072,164,948

12742 / 79,072,164,948 = 0.16114

Which gives me 0.16114mm

Q2:
76,700,000,000,000 / 12742 = 6,019,463,193

42.67 x 6,019,463,193 = 256,850,494,428

Which gives me 256,850KM

 

[Ian H]

Mike,
Ian says answer to Q1 (New Dia of Earth) is 0.1485mm and Rsgaz and I said 0.161mm, so close enough to be the same as we are using slightly different distances for distance of Sirius.
SFK

I put 12472 instead of 12742 for the size of the earth.

I blame my phone, I had to turn it sideways to get the big numbers to show

 

[SFK]

Mike,
If you are actually asking why
"rays of light from the stars (Sirius for example) are parallel when they arrive at Earth irrespective of where we are bearing in mind the Earths diameter 12,742km"

I see that being because the emission of light from the ball of Sirius is at all angles.
But as we are so so far away it is like a triangle with one impossibly long side (the distance between Earth and Sirius) and one short side (the diameter of earth).
When you calculate the angle then you get near zero.

sinθ =o/h
θ = sin-1 (o/h)
θ = sin-1 (12742/76,700,000,000,000)
θ = sin-1 (12742/76.7E12)
θ = sin-1* 1.66E-10
θ = 0.000000009 degrees
θ = near zero degrees

 

[ban-all-sheds]

If I park my car at the end of the road it looks smaller but it’s really the same size.

 

[EFLImpudence]

" rays of light from the stars (Sirius for example) are parallel when they arrive at Earth

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

 

[SFK]

ELFImpudance ,
Yes the light is converging from the Sun and also from Sirius (hence my triangle maths),
They are converging from the Sun (and due to proximity and size of sun they are also diverging and parallel) and this convergence can be measured.
But the tiny convergent angle of the rays from Sirius is so small that the light from Sirius can be said to be parallel when using (typical) earth optics/telescopes.

However, with interferometry, the tiny angles of the converging rays from Sirius can be measured to calculate the angular size of Sirius: https://en.wikipedia.org/wiki/Hanbury_Brown_and_Twiss_effect

SFK

 

[durhamplumber]

I still don’t get it. A KM is 1000mtrs wether your measuring from a bus in London to bus in Manchester or from a golf ball in Spain to a Golf ball in Scotland.

It is to shrink things down to become measurable,so a map is to scale,,so a 1:1000 map is metre across and not a km,,if it shows a km.

 

[durhamplumber]

Hi Ian & SKF,

I note you do not agree on the distance to Sirius

76,700 trillion
81,460 trillion

Mike

Tiny difference percentage wise,depends how precise you are about speed of light,to how many decimal places rounding up is too.

 

[wobs]

I'm expecting adverts for Sirius Cybernetics Corporation across the top of the page.

"Share & Enjoy"

 

[Ian H]

It is to shrink things down to become measurable,so a map is to scale,,so a 1:1000 map is metre across and not a km,,if it shows a km.

Yes, I get that now but I read the original question differently when I rolled out of bed this morning.