Converting Astronomical Objects to Money: Earth, Sun & Milky Way

[Freeman Dyson]

Homework Statement

Convert astronomical objects into money based on their radius. This is how it works. The Earth is worth one penny. $.01 The Earth's radius is 6,378 km. So 6,378 km equal one penny in this system. So the sun, which has a radius of 696,000 km. We divide that by 6,378. Which is 109 cents. So the sun is worth 1.09

Solar System is 40 AU. I want this in km. AU is 1.5 x 10^8 km. So do 40 times 1.5 x 10^8 =6,000,000,000km or 6 x 10^9. So then I divide 6 x 10^9 by 6.378 x 10^3. And I get 940,733 cents. Drop 2 decimals to convert to dollars and I get $9,407 for the price of the solar system.


I believe those are right. Anyway, it is the bigger ones that I am having problems with. Next is the Milky Way. Milky way is 50,000 light years away.

Homework Equations

The Attempt at a Solution

So this is how I am going about the Milky Way one. Probably wrongly. I want to convert the those 50,000 light years into km. Or do I? Anyway, I'll try it. A light year is 10 trillion km. So I multiply 50,000 by 10 trillion. 5 x 10^4 x 1 x 10^13. So that is 5 x 10^17. Now I divide this by 6.378 x 10^3 and get .78394481 x 10^14. Now what? Should I get rid of some exponents? Maybe 783 x 10^9? Which is 78,300,000,000,000 cents. Take out two points to convert to dollars and you have $783,000,000,000 as the cost of the Milky Way, Is this correct? Is there a better way to do it?

 

[Mentallic]

I guess you're trying to create an analogy of the size of the universe in terms of Earth's radius.

First of all, 0.78394481 x 10^14\approx784x10^10 so that makes $7.84 trillion.

And what do you mean by the Milky Way is 50,000 light years away? We are IN the milky way and as a note its radius is approximately 50,000 LY.

The way you're doing the calculations is fine. But just to neaten up your work a bit, if you're going to approximate the light year so much so that you say it's 10 trillion km (while 9.46 trillion is a closer approximation) then you should only stick to 2 or 3 significant figures at most.

.78394481 x 10^14 is unnecessary

EDIT: sorry I forgot the 7.84 x 10^12 was in cents. The answer is $78.4 billion.

 

[Freeman Dyson]

I guess you're trying to create an analogy of the size of the universe in terms of Earth's radius.

First of all, 0.78394481 x 10^14\approx784x10^10 so that makes $7.84 trillion.

And what do you mean by the Milky Way is 50,000 light years away? We are IN the milky way and as a note its radius is approximately 50,000 LY.

The way you're doing the calculations is fine. But just to neaten up your work a bit, if you're going to approximate the light year so much so that you say it's 10 trillion km (while 9.46 trillion is a closer approximation) then you should only stick to 2 or 3 significant figures at most.

.78394481 x 10^14 is unnecessary

Thanks. I meant the radius of the milky way is 50,0000 ly. I had a feeling that .78394481 was a little ridiculous. So would .783 be fine? And where are you getting 784 x 10^10 from? I get 783 x 10^10. Or 78.3 x 10^12 divided by 10^2 for dollars = 78.3 x 10^10. So now I am actually getting 783 billion.

 

[Freeman Dyson]

Anyone check my work on this?

milky way=50,000 ly across. 10 trillion km in a light year, so:

5 x 10^4 x 1 x 10^13 = 5 x 10^17.

I must divide that by the Earth's radius to find out how many cents. So, 5 x 10^17 divided by 6.378 x 10^3 =.783 x 10^14

I convert that to 78.3 x 10^12 for the total cents. Now to convert that to dollars I divide 78.3 X 10^12 by 10^2. So I get 78.3 x 10^10 dollars. Which is 783 billion.

 

[Mentallic]

I had a feeling that .78394481 was a little ridiculous. So would .783 be fine?

Actually since you rounded the light year to 1 significant figure, it would be more acceptable to leave it as 0.7 x 10^14.

And where are you getting 784 x 10^10 from? I get 783 x 10^10.

I just rounded off. 0.7839... rounded to 3 decimal places is 0.784

Or 78.3 x 10^12 divided by 10^2 for dollars = 78.3 x 10^10. So now I am actually getting 783 billion.

No no. 783 x 10^10 = 78.3 x 10^11 = 7.83 x 10^12 and then dividing this by 100 to make it into dollars is 7.83 x 10^10

There isn't much point in trying to be too much more accurate than this, even with the light year because the Milky Way doesn't have an outside "wall" that separate the inside and outside of the galaxy, so this is why 50,000 LY for its radius is an approximation.

 

[Freeman Dyson]

Actually since you rounded the light year to 1 significant figure, it would be more acceptable to leave it as 0.7 x 10^14.
I just rounded off. 0.7839... rounded to 3 decimal places is 0.784

No no. 783 x 10^10 = 78.3 x 10^11 = 7.83 x 10^12 and then dividing this by 100 to make it into dollars is 7.83 x 10^10

There isn't much point in trying to be too much more accurate than this, even with the light year because the Milky Way doesn't have an outside "wall" that separate the inside and outside of the galaxy, so this is why 50,000 LY for its radius is an approximation.

thanks.

Where did I go wrong? 5x 10^17 divided by 6.378 x 10^3 is .783 x 10^14. I move the decimal 3 places and get 783 x 10^11. Where do you get 783 x 10^10?

ah i missed a decimal place up there. I moved 2 instead of one. It should be 78.3 x 10^11. I had it at 78.3 x 10^12. I get 78.3 billion. you were right from the start

 

[Mentallic]

thanks.

Where did I go wrong? 5x 10^17 divided by 6.378 x 10^3 is .783 x 10^14. I move the decimal 3 places and get 783 x 10^11. Where do you get 783 x 10^10?

ah i missed a decimal place up there. I moved 2 instead of one. It should be 78.3 x 10^11. I had it at 78.3 x 10^12. I get 78.3 billion. you were right from the start

Looks like you found the problem
I'm glad to help. But may I ask what this was for? Curiosity? Or are you trying to give an analogy of how large our galaxy is in terms people can somewhat comprehend?

 

[Freeman Dyson]

Looks like you found the problem
I'm glad to help. But may I ask what this was for? Curiosity? Or are you trying to give an analogy of how large our galaxy is in terms people can somewhat comprehend?

Yes, he is trying to show us just the immense scale of the universe. So we can see that our own solar system is like $9,000. The Milky Way would be 78 billion, etc..

 

[Mentallic]

And does the $80 billion make the galaxy seem big to you?
I've already accepted that I wouldn't be able to comprehend how big it really is. I can't even come to terms with how big, yet too small, Earth is.

 

[Freeman Dyson]

I already knew the universe was huge. For the Local Group I got 2.5 trillion. And for the visible universe I got 215 quadrillion.

 

[Mentallic]

Yes I know it's huge too, and this is also what anyone would tell you. The only difference is that when I try to comprehend how huge it is, using analogies in a similar fashion as you have still doesn't help me. For e.g. The Earth is already so big, but to take that as being 1 cent and having the visible universe costing 215 quadrillion doesn't help whatsoever since that sum of money is much too big for me to understand.

Another analogy: let the radius of the Earth be the radius of a typical hydrogen atom (10^-10 m) then the galaxy is 800m which is a distance I can come to grips with obviously, but the only problem is that I can't fathom the size of the atom. The visible universe will allow you to circumnavigate the globe 50 times!
I always get either one case where the starting size of say 1km or the globe's diameter is a small but understandable quantity, but then comparing this to the universe the quantity becomes much too big, or, the starting quantity is much too small to comprehend but the outcome I could fathom.