18 1. TOOLS FOR UNDERSTANDING SPACE
Table 1.3: Scaling ratios for sizes and distances of Earth, Moon, and Sun
R
⊕
6.380 × 10
6
m
R
⊕
/R
3.67
R
1.738 × 10
6
m
R
/R
⊕
109
R
6.963 × 10
8
m
d /R
⊕
60.2
d
3.84 × 10
8
m
d
/R
215
d
1.496 × 10
11
m
d
/R
⊕
23,400
Table 1.3 shows some examples of these scaling ratios. Note that the ratios are (and should
be) dimensionless, and this requires that both lengths be expressed in the same units before
dividing one by the other.
Scaling ratios show us important relationships at a glance. e Sun has 109 times the
diameter of Earth. e Moon is at a distance of 60 Earth-radii (30 Earths side-by-side). e
distance from Earth to Sun is roughly the same as 100 suns—or 10,000 Earths—placed side-
by-side.
1.4 SURFACE AREA, VOLUME, MASS, AND DENSITY
An official NBA basketball should have a circumference of 29.5 inches, or 74.9 cm, for a diameter
of 74:9= D 23:9 cm. An official ITF tennis ball, on the other hand, should have a diameter
between 6.54 and 6.86 cm. is means a basketball has 23:9=6:54 D 3:65 times the diameter of
a smaller-sized tennis ball. By an odd coincidence, this is almost exactly the size of the Moon as
compared to Earth; the diameter of Earth is 3.67 times the diameter of the Moon (see Table 1.3).
Simply over-inflate the basketball slightly, and the comparison to the Earth and Moon could be
made exact.
Place a basketball next to a tennis ball (see Figure 1.9) and one could be forgiven for
thinking the basketball is considerably more than 3.7 times bigger than the tennis ball. e
reason is that when we visually compare spherical objects to each other we seem to do so more
in terms of their surface areas than their diameters [see, for example, Jansen and Hornbak, 2016].
is suggests we must be careful when we say, for example, that one object is 3.68 times “bigger”
than another. Do we mean its diameter, surface area, or volume?
For spherical objects, the relations between diameter (or radius, R D diameter=2), surface
area, A, and volume, V , are simple:
A D 4R
2
(1.31)
V D
4
=3R
3
: (1.32)
1.4. SURFACE AREA, VOLUME, MASS, AND DENSITY 19
Figure 1.9: e diameter of a tennis ball as compared to a basketball is nearly the same ratio as
that of the Moon compared to Earth.
If we mean to only compare these quantities for two different objects, it is simpler still:
A
1
A
2
D
R
1
R
2
2
(1.33)
V
1
V
2
D
R
1
R
2
3
: (1.34)
Although Equations (1.31) and (1.32) are valid only for spheres, Equations (1.33) and (1.34)
are correct whenever one compares any two objects, so long as they both have the same shape.
Since surface area scales with radius squared, its SI units are not meters (m), but rather square
meters (m
2
). And it is also clear that volume must be measured not in meters, but rather in cubic
meters (m
3
).
To say that the radius (or diameter) of Earth is 3.68 times the radius of the Moon is to say
that Earth’s surface area is
3:68
2
D
13:5
times as great, while its volume is
3:68
3
D
49:8
times
greater. And so although a basketball has only 3.65 times the diameter of a tennis ball, it has
over 13 times the surface area, and it is this larger number that is more connected to our human
perception when we visually compare the “sizes” of objects.
e surface area of a spherical astronomical body has an important physical significance;
it is related to the (perhaps invisible) radiation the body emits. But the volume is also important;
it tells us something about the total amount of stuff the body is made of. e amount of water
needed to fill a hollow spherical container, for example, is directly proportional to the container’s
volume rather than its surface area or radius.
20 1. TOOLS FOR UNDERSTANDING SPACE
But there is another important physical quantity that is related to (but not the same as)
volume—the object’s mass. We will consider mass often throughout e Big Picture; it is a subtle
and complex subject. But as a start, let us consider these two different and very-crude definitions
of mass.
1. On the surface of Earth, an object with more mass weighs more than an object with less
mass.
2. An object with greater mass causes your foot to hurt more when you kick it, compared to
kicking an object with less mass.
is first definition is clearly about gravity, and in this crude and simple form it is not of much
help; how does one “weigh” the Moon? e second definition is called inertia, and it seems to
be distinct from gravity. Bring a cathedral with you to a blank region of interstellar space—so
gravity is an insignificant factor—and then kick it. Your foot will hurt just as much as if you had
kicked that same cathedral here on Earth. We will make both of these definitions of mass more
precise as we go along. And eventually we shall see that these two seemingly distinct ways to
look at mass are not so different as they seem.
Clearly, mass has something to do with the total amount of “stuff” an object is made
of. And so there is a connection between mass and volume. If one lead balloon
9
has twice the
volume as another, it also has twice the mass. But of course it is not only about volume; a lead
balloon would have far more mass than an ordinary air-filled balloon of the same volume. And
so there is some property of the type of material itself—lead vs. air, for example—that is also
connected to the mass of an object. at property is density,
10
and we define its average value
for some object in this way:
N D
m
V
; (1.35)
where m is the object’s mass and V is its volume, and we by tradition employ the Greek letter
(rho) as the symbol for density, and N for its average value.
Equation (1.35) is only the average density of the object (we place a bar over a variable
to represent its average value); it may be made of different materials, with some parts of higher
density and others of lower density. But one can take any arbitrary piece of an object and use
Equation (1.35) to calculate the average density of that part. If one chooses a tiny-enough part,
such that the material throughout is all the same, then the calculation represents the density, ,
of the material at that particular point in the object.
e average density of an object provides important clues regarding the type of material
it is made of. On the other hand, if the composition of the object (and thus its density) can be
guessed, we can then measure its volume and rearrange Equation (1.35) to calculate its mass:
m D NV: (1.36)
9
I should make some sort of humorous comment at this point, but I fear it would go over like ….
10
My jokester father has been known to say that an object of high density is “heavy for its weight.”
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