The Scale of our Solar System
Most books and web sites are a bit misleading when it comes to scale our solar system. This is because they are usually confined by space on the page or screen. Your job is to bust that myth and set things straight. Imagine the entire solar system was shrunk down and all you knew was that the sun (on this shrunken model) was the size of a standard basketball. With that, please calculate (showing all logic and steps):
You will need to look up some things. Here are some numbers you will need to know:
Diameter of a basketball (DBB) = 23.87 cm (approximately)
Diameter of the real sun (DSun) = 1,391,684 km
1. How large would the Earth be on this shrunken model?
If you know that the sun is 109 times larger than the earth, all you need to do is divide the diameter of the basketball by 109: DBB/109 = 23.87 cm/109 = .22 cm
If you didn't know that fact, all you need to do is look up the diameter of the earth (DEarth) which is 12,742 km ... and divide DSun by DEarth : DSun/DEarth = 1,391,684 km/12,742 km = 109
Another approach is to set up a proportion:
DSun
DBB
_____ = ______
DEarth X where X is the diameter of the shrunken Earth
2. How far away from the basketball would it orbit?
You will need to know the real Earth-Sun distance (ESdistance) = 149,600,000 km
Now set up a proportion that compares values in the real universe to comparable values in our shrunken model. It is not necessary to convert all values to similar units ... just as long as you use compare things the same way. For example, here is one way you can set up the proportion:
DSun
DBB
________ = _____
ESdistance X
where X is the Earth-Sun distance in the shrunken model in the same units as the
diameter of the basketball
1,391,684 km / 149,600,000 km = 23.87 cm / X but now X must be measured in cm
X = 2,568 cm .... which should be converted to 25.68 meters
3. How big would you be?
I am 6 feet tall ... We can use the same approach as the last answer to set up a proportion but it is important that the unit comparison matches. We can do this a couple of ways
DSun
DBB
_______ = _______
My Height X where X is my model height
OR
DSun
My Height
_____ = _________
DBB X
.... either way it is important to convert the units so the comparison is consistent
I will use this second proportion and compare the diameters (both now in km) and also covert my height to centimeters so that the value for X will also be in centimeters. 6 feet = 183 cm and 23.87 cm = .0002387 km
1,391,684 km
183 cm
___________ = _______
.0002387 km X cm
X = .000000031 cm ...... which is a very small number. An appropriate answer would be that your height is .31 nanometers or 3.1 Å (angstroms) or 310 picometers. This is slightly larger than an atom of uranium.
4. How big would Jupiter (our largest planet) be?
Just repeat the steps found in question 1. Jupiter's diameter is 139,822 km ... or just find out that Jupiter is about 11 times larger than the earth. You will get a model Jupiter to be about 2.4 cm across ... about the size of a quarter.
5. How far away from the basketball would you find Pluto?
Pluto has a pretty eccentric orbit but it averages about 40 AU from the sun. Simply multiply your answer to question 2 by 40 to get an approximate distance of about 1 km.
Bonus points: How far away from the basketball would the closest star be on this shrunken model?
The closest star (Proxima Centauri) is 4.22 light years from the sun. Let's convert this to 3.99 x 10 13 km. Sorry but you really need to do this with scientific notation or you will go batty with all those zeros. Here is an appropriate proportion:
DSun
DBB
____ =
______
DStar X
Where DStar is the distance to the closest star .... converted to km and X is the model distance to the star in the same units as the diameter of the basketball (cm)
1,391,684 km
23.87 cm
___________ = ________
3.99 x 10 13 km X
X = 6.84 x 10 8 cm which converts to 6840 km or 4250 miles ..... now that is pretty far away even on this scale!!!