# Scalling the Universe

Sizes of the Planets

You have read about the planets in our solar system many times. However, your readings may not have given you an appropriate idea of the vast space within our solar system.  This project will try to do that.  Obviously we cannot make an exact replica of the Sun and planets, so instead we must scale everything down to a more approachable size.  Just as a map uses a scale to represent distances, in this project we will also use a scale… the scale that we will be using for this lab is to approximate the Sun with a beach ball, or every 1,338,000 km will be equal to 40 cm.  Using this scale and ratios, you should be able to determine the scaled diameters of the Solar System objects.

Scaling Factor  =  scaled size / real size =  $2.9895*10^{-10}$

Fill in the appropriate scaled diameters for the objects in the chart below.  Show at least one sample calculation below the table.

Object                          Mean Diameter(km)                 Scaled Diameter(scaled cm)
Sun                                1,338,000                                  40
Mercury                         4,900                                         0.15
Venus                             12,100                                       0.36
Earth                              12,800                                       0.38
Earth’s Moon                 3,500                                         0.10
Mars                               6,800                                         0.20
Jupiter                            143,000                                     4.28
Saturn                            120,000                                      3.59
Uranus                           51,000                                       1.52
Neptune                         50,000                                       1.49

Show work here:
Earth scaled diameter (m) = Earth real diameter (m) * scaling factor = $12800000*2.9895*10^{-10} =0.0038 m =0.38 cm$

How long is your commute to school in real kilometers?  Now how long is your commute to school in our scaled distance? How tall are you on this scale? Compare your scaled size to a real object (ie “I would be the height of a grain of sand). Show all your work! Attach additional sheet if needed.

My commute to school in real km is 7 km. In the above scaled distance my commute to school is

$7000*2.9895*10^{-10} =2.092*10^{-6} m = 2.1 \mu m$. It is about one tenth to one hundred of the diameter of a human hair.

My height is 1.75 m on the real scale. In the above scaled distance my height is $1.75*2.9895*10^{-10} = 4.48*10^{-10}$.

It is about the wavelength of X rays.

Distances of the Planets from the Sun:

You will now make a scale model of the distances of the planets from the Sun.  Using the same scale, fill in the scaled distances from each planet to the Sun in cm.  Also convert these scaled centimeters into scaled km.  You will not apply the scale factor again, you need to convert.

Object Mean Distance from the Sun (km)Scaled distance from the Sun (cm)Scaled distance from Sun(km)

Mercury          58,000,000                                          1734                                   0.017
Venus              108,000,000                                         3229                                  0.032
Earth               150,000,000                                         4484                                  0.045
Mars                228,000,000                                         6816                                 0.068
Jupiter             778,000,000                                         23259                                0.233
Saturn              1,430,000,000                                     42750                                0.428
Uranus             2,870,000,000                                     85800                                0.858
Neptune           4,500,000,000                                     134529                              1.345

Suppose that you want to figure out how far away the nearest star system is using the same scale you used above.  The Alpha Centauri system is the closest, and it is 4.4 light-years (ly) away from us.  Knowing the following conversions, figure out how far away (in scaled centimeters and scaled kilometers) the Alpha Centauri star system would be; also, compare your result to the actual dimensions of some real object, i.e. on this scale the nearest star system would be as far away as Conway. Show all your work.
$1 ly = 9.46 x10^{12} km$         $1 km = 100000 cm$
Real distance = $4.4*9.46*10^{12} =4.16*10^{13} km$

Scaled distance (m) = Real distance (m) * scalling factor = =$4.16*10^{16}*2.99*10^{-10}=1.244*10{^7} m =1.244*10^9 cm =1.244*10^4 km$

This distance is comparable to the medium width of the Pacific ocean.

How big (in scaled km) is the Milky Way in this scale? Use 100,000 ly for the actual width of the Milky Way. What  is about this big in our universe (ie if the sun were a beach ball, the Milky Way would be the size of …)?
Real diameter of galaxy = $10^5 ly =10^5*9.46*10^{12} =9.46*10^{17} km$

Scalled diameter of galaxy (km) = $2.83*10^8 km$

This diameter is comparable to the distance between Earth and Mars.

How far away is the Andromeda Galaxy on our scale?  Look up its actual distance and convert to scaled km.  Compare this scaled distance to some real distance (i.e. if the Sun were a beach ball, the AG would be the same width as here to

The distance from Earth to Andromeda Galaxy is $2538000 ly =2.538*10^6 ly=$ $=2.538*10^6*9.46*10^{12} =2.4*10^{19} km$

Scaled distance (km) = $7.18*10^9 km$

This distance is about twice the distance from Earth to Neptune

Scale Model of the Milky Way

The Milky Way galaxy is a large spiral shaped group of stars that includes our Sun.  Our Sun is positioned about two-thirds of the way out from the center of our galaxy.  This next scale model will help you visualize the size of the Milky Way, and our location in the galaxy.  For this activity, we will use a scale where 1 mm represents 500 light-years.

Scaling factor = $1 mm/500 ly = (1/500) mm/ly = 10^{-3}/500/9.46*10^{12} =2.11*10^{-19}$

1. Calculate the diameter of the disk of the Milky Way in this scale. (The approximate diameter of our galaxy is 100,000 light-years.) Draw a circle of this diameter on a piece of paper and turn it in with the report. (Show your work.)

Real diameter = $100000 ly$

Scaled diameter = $200 mm$

2. The thickness of the disk of the Milky Way is around 3000 light-years. In our scale, how thick is the Milky Way?

Real thickness =$3000 ly$

Scaled thickness = $6 mm$

3. The galaxy has a bulge at its center. The bulge is spherical in shape with a diameter of 10,000 light-years. How big is the bulge in our scale? Draw a circle to represent the bulge at the appropriate location on your drawing.

Real diameter = $10000 ly$

Scaled diameter = $20 mm$

4. Calculate the diameter of the solar system in this scale. The diameter of our solar system out to the Oort cloud is ~1 ly. Show your work. Draw a circle representing the Solar System on your scaled down galaxy at the appropriate location; will you be able to see it? Can you draw the Sun on this scale?

Real diameter = $1 ly$

Scaled diameter =$0.002 mm$

5. The Andromeda Galaxy is almost the same size as our galaxy. It is the closest spiral galaxy to the Milky Way: only about 2,300,000 light-years away. At this scale, how far part are these two galaxies?

Real distance = $2300000 ly$

Scaled distance = $4600 mm$

Visualize the model of the Sun as the beach ball, and the scale distance separating it and Alpha Centauri, the nearest star.  Compare this to the scale size of the Milky Way (500 ly = 1 mm) and the scale distance between it and the nearest spiral galaxy.  Which are closer relative to their sizes: stars or galaxies?  To do this, calculate the number of stellar radii between stars and the number of galactic radii between galaxies on their respective scales.  The smaller of these numbers will tell which objects are relatively closer.   Show all your work.
The rapport of the two scales is = $(2.985*10^{-10})/(2.11*10^{-19}) = 1.4*10^9$

The diameter of the sun is $1,338,000 km$

The distance from the sun to the nearest star (Proxima Centauri) is $39,900,000,000,000 km$

The number of stellar Radii between stars = $2.98*10^7$

The diameter of the Milky way = $100000 ly$

The distance to the nearest galaxy = $2300000 ly$

The number of galactic radii between galaxies = $23$

The galaxies are relatively closer one to each other than the stars