ASTRO 1050
Scientific Notation, Model Scales, and Calculations
The simple truth is, interstellar distances will not fit into the human imagination. - Douglas Adams
Materials: Ping pong balls, meter sticks, protractors
1. Scientific Notation
Describing the universe requires some very big (and some very small) numbers. Such numbers are tough to
write in long decimal notation, so we’ll be using scientific notation. Scientific notation is written as a power
of 10 in the form
m x 10e
where m is the mantissa and e is the exponent. The mantissa is a decimal number between 1.0 and 9.999 and
the exponent is an integer. To write numbers in scientific notation, move the decimal until only one digit
appears to the left of the decimal. Count the number of places the decimal was moved and place that number
in the exponent. For example,
540,000 = 5.4x105 or, in many calculators and computer programs this is written: 5.4E5 meaning 5.4
with the decimal moved 5 places to the right. Similarly
314.15 = 3.1415x102 and
0.00042 = 4.2x10-4
and
234.5x102 = 2.345x104
You get the idea. Now try it. Convert the following to scientific notation.
Decimal
Scientific
Decimal
Scientific
2345.4578
__________________
0.000005
__________________
356,000,000,000
__________________
0.0345
__________________
111x105
__________________
2345x10-8
__________________
2. Arithmetic in Scientific Notation
To multiply numbers in scientific notation, first multiply the mantissas and then add the exponents. For
example, 2.5x106 x 2.0x104 = (2.5x2.0) x106+4 = 5.0x1010 . To divide, divide the mantissas and then subtract
the exponents. For example, 6.4x105 / 3.2x102 = (6.4 / 3.2) x 105-2 = 2.0x103
Now try the following.
4.52x1012 x 1.5x1016 = ______________________
9.9x107 x 8.0x102 = ______________________
1.5x10-3 x 1.5x102 = ______________________
8.1x10-5 x 1.5x10-6 = ______________________
1.5x1032 / 3.0x102 = ______________________
8.0x10-5 / 2.0x10-6 = ______________________
Be careful if you need to add or subtract numbers in scientific notation.
4.0x106 + 2.0x105 = 4.2x106 since
4.0x106 =
4,000,000
5
+2.0x10 =
+ 200,000
4.2x106 =
4,200,000
3. Converting Units
Often we make a measurement in one unit (such as meters) but some other unit is desired for a computation or
answer (such as kilometers).
Example: You have 2,340,000,000,000 meters. How many kilometers is this? How many astronomical units?
There are 1000 m/km. Because kilometers are larger than meters, we need fewer of them to specify the same
distance, so divide the number of meters by the number of meters per kilometer and notice how the units cancel
out and leave you with the desired result.
2.34x1012 m
9
= 2.34× 10 km
m
Another way to think about this operation, is that you want fewer km than m,
1000
km
so just move the decimal place three to the left since there are 103 m per kilometer. Or, if the new desired unit
is smaller, and you expect more of them, then multiply. For example, how many cm are there in 42 km?
cm
42 km× 105
= 42× 105 cm
km
Use the information in the appendix of your text to convert the following. Write these in Scientific Notation.
50 km =
_____________________ m 3x106 m =
____________________ cm
52,600,000,000 km = _____________________ m
300,000,000,000 km = ____________________ AU
6.0x1018 m =
_____________________ ly
5.2x1012 kg =
_____________________ g
3450 seconds =
_____________________ minute 99 minutes =
____________________ hr
600 hours =
_____________________ days
_____________________ yr
1 year =
_____________________ s
4500 parsecs =
____________________ km
4.0x10-5 sun masses = ____________________ kg
1200 days =
4. Scale Models
The Universe is such a big place, scale models are almost required to help visualize it and help it fit into our
imagination. If we are going to represent the earth by a ping long ball (diameter=4.0 cm), how large should be
the planet Venus? To make the scale model we need to look up the real size of Earth and Venus and use the
method of ratios to find the model size of Venus.
12756 km real Earth diameter = 4 cm model size
12101 km real Venus diameter = x cm model size
Set up the ratio, and then solve for x (cross multiply and divide)
12101km
12756 km 4cm
x cm= 4cm×
= 3.79cm
=
therefore,
12756km
12101km x cm
Now do the following:
Real Size (km)
The moon (diameter)
The distance to the moon
Model size (cm)
The distance from sun to earth
The sun (diameter)
The size of Jupiter (diameter)
The radius of Pluto’s orbit (semi-major axis)
Use the space below for calculations.
5. Time to travel
Suppose you could travel at the speed of the fastest jets, about 1500 miles per hour, or about 2250 km/hr. The
distance, D, traveled in time, t, and speed, v, is
D(km) = v(km/hr) x t(hr)
D(km) = v(km/s) x t(s)
D(m) = v(m/s) x t(s)
Notice that the units of time in v and in t must be the same, as must the units of distance in the v and in D,
otherwise, we’ll get a nonsensical answer!
Traveling at the speed of a fast jet, how long would it take to get from earth to:
Distance (km)
Time
The moon:
_____________________
_________________
sec
The sun:
_____________________
_______________
hours
Mars (closest to Earth)_____________________
___________________days
Alpha Centauri:
_____________________
____________________days
Now suppose you could travel at the speed of light (3x108 meters/sec). How long would it take to get to:
Distance (km)
The sun:
Time
_____________________
Pluto (closest to Earth):_____________________
Alpha Centauri:
_____________________
____________________minutes
____________________hours
____________________years
6. Angles
Often in astronomy we measure the size of things in the sky, or the motion of things across the sky, in terms of
angles. In one circle there are 360 degrees. In one-half of a circle there are 180 degrees, etc.
90°
180°
30°
Make a rough estimation of the following angles. Then use a protractor to measure them.
_______________ degrees
_______________ degrees
_______________ degrees
Degrees are further subdivided into “minutes of arc” or arcminutes. There are 60 arcminutes in one degree.
Acrminutes are further subdivided into “seconds of arc” or arcseconds. As you might imagine, there are 60
arcseconds in one arcminute. Convert the following.
120 arcminutes = __________________ degrees
900 arcseconds = _________________ arcmin
12 arcminutes = __________________ degrees
6 arcseconds = _________________ degrees
12 degrees = __________________ arcminutes
360 degrees = __________________ arcmin
57.29 degrees = _________________ arcsec
55 arcmin =
_________________ arcsec
360 degrees = _________________ arcsec
This last one is special. This number will show up a lot this
semester. And here’s where it comes from.
There is also a special unit of angular measure called a radian. 1 radian = 57.3 degrees.
We won’t use it explicitly this semester, but we’ll be using it implicitly,
so we thought you might like to know.
1 rad = 57.3 deg
7. Measurements and Graphing
Measure the height and shoe length (we really mean foot length, but we don’t want everyone taking off their
shoes in lab....) and the length of fingernails of your lab mates, then record data for everyone else in your lab.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
Height
(cm)
________
_______
_______
_______
_______
_______
_______
_______
_______
_______
Shoe Length Nail Length
(cm)
(mm)
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
_______ _______
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
Height
Shoe Length
(cm)
(cm)
________ _______
________ _______
________ _______
________ _______
________ _______
________ _______
________ _______
________ _______
________
_______
________ _______
Nail Length
(mm)
_______
_______
_______
_______
_______
_______
_______
_______
_______
_______
Do you expect to see a correlation between height and shoe length? Between shoe length and nail length?
Why or why not?
Now plot these data on the graphs provided. You’ll need to add the axis numbers.
Nail length (mm)
Shoe length (cm)
140 145 150
155
160
165 170 175 180
Height (cm)
Which variables show correlations?
What is the explanation for this correlation?
185
190
140 145 150
155
160
165 170 175 180
Height (cm)
185
190