Some Math

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Some Math
Algebra D vt, and other formulae
Slope of a curve, line
Inverse Square Laws, e,g, F Useful
Concepts
DY RISE DX RUN
GMm
R2
Measuring Circles and arcs Scientific
Notation  Système Internationale, the Metric
System
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Cardinal Points
N
NE
NW
E
W
SE
SW
S
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Cardinal Points
N
NNE
NNW
NE
NW
ENE
WNW
E
W
ESE
WSW
SE
SW
SSE
SSW
S
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Angles Why are they convenient?
How do we measure them?
If we see two objects very far away - how can we
describe the distance between them without
knowing how far away they are? How can we
describe the distance between them?
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90
270
360
180
Measure an arbitrary angle with a protractor
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Angles in Astronomy
C
A
q
q
B
D
Do we carry around protractors? Yes, our fists
10 at arms length,
We say the distance between the stars subtends an
angle q
We measure angles in degrees, 360 in a full
circle arc
minutes, 60, in 1
arc seconds, 60, in 1
How many arc seconds are there in a full circle?
360
X 60
X 60
12,960,000 arc seconds
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Angles in Astronomy
Altitude, height of an object above your horizon.
The zenith has an altitude of 90.
Azimuth - measured in degrees from the north,
eastward.
North 0 azimuth East 90
azimuth, South 180
In the Celestial Globe Lab we will determine
where a planet or star rises. The position on
the horizon is given by its azimuth. N 0, E
90, S 180,
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Locating Places, Stars,
e.g. Stars
on Earth
By Name Relative Location Absolute Location
Rigel for example
Philadelphia
altitude and azimuth
80 SW of NYC
Latitude and Longitude
Right Ascension and Declination
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Celestial Globe
North Celestial Pole
Summer Solstice
Vernal
Celestial Equator
Equinox
Ecliptic
Latitude
-gt Declination
Longitude
-gt Right Ascension (only eastward)
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Scientific Notation
Powers of 10
1,000,000,000,000 1012 one trillion
tera 100,000,000,000 1011
10,000,000,000 1010 1,000,000,000
109 one billion giga
100,000,000 108 10,000,000
107 1,000,000 106
one million mega 100,000
105 10,000 104
1,000 103 one
thousand kilo 100
102 10 101 1 100
one 10-1 .1 deci 10-2
.01 centi 10-3
.001 milli 10-4 .0001 10-5
.00001 10-6
.000001 micro 10-7
.0000001 10-8 .00000001 10-9
.000000001 nano 10-10
.0000000001 10-11
.00000000001 pico 10-12
.000000000001
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Scientific Notation
In astronomy we deal with the very large and the
very small. To write these numbers with
excessive zeroes 5.2 Au x 150,000,000 km/AU
780,000,000 km We write the number as a number
between 1 and 10 x the
appropriate power of 10 780,000,000 km 7.8 x
108 km G the gravitational constant
.0000000000667 Nm2/kg2
6.67 x 10-11 Nm2/kg2
Move the decimal point of the original number
so that it is to the right of the first
non-zero digit. Rewrite the number with the
appropriate number of digits. Count the number
of places you moved the decimal point. This is
the exponent of 10. But - If you moved
the decimal point to the right, i.e. the number
was less than 1, then place a negative on
the exponent. - If you moved the decimal
point to the left, I.e. the number was greater
than 1 then use the exponent without a
sign. (A positive exponent is then implied.)
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Write 3,837.552 in Scientific Notation
Write the number as a number between 1 and 10
times the appropriate power of 10 1 - Place
a decimal point after the first non-zero number
3.837552
2 - The decimal point has been moved 3 places to
the left. So the appropriate power of 10 is
3 or 103.
3.837552 x 103 If we round to 2 places 3.84 x
103
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Write .00000893741 in Scientific Notation
Write the number as a number between 1 and 10
times the appropriate power of 10 1 - Place
a decimal point after the first non-zero number
8.93741
2 - The decimal point has been moved 6 places to
the right. So the appropriate power of 10
is -6 or 10-6.
8.93741x 10-6 If we round to 2 places 8.94 x
10-6
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Système International dUnités The
Metric System (MKS System)
Length Meter Mass Kilogram Time Second Te
mperature Kelvin Electric Current Ampere

• 1000 meters I kilometer
• 1 meter 100 centimeters
• 1 meter 1,000,000,000 nano meters
• 2.54 centimeters 1 inch
• 1.6 kilometers 1 mile

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Scale
Real World Value
Scale
Measured Value
Moons real world diameter is 2,160 miles In
this diagram the scale Scale
360 miles/cm or we say 1 cm 360 miles

2,160 miles
6 cm
6 cm
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Pythagorean Theorem
2
2
2
C
A
B
B
A
B
Compare the area of the
C
outer square, side A B,
A
C
with the area of the inner
square, side C, the
area of each of the right
C
A
triangles, side A and B to
C
show that the Pythagorean
B
Theorem holds.
B
A
We will use the Pythagorean Theorem in the
Spectra Lab
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Formulas for the Spectra Lab
C
B
B
A set to 30 cm
Wavelength(Å) 15,419xB Å
C
2
C 900 B
and then calculate the error
Å Angstrom 1/10,000,000,000 of a meter or
10-10 m
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Formulas for the Spectra Lab
C
B
B
A set to 30 cm
Wavelength(nm) 1541.9xB nm
C
2
C 900 B
and then calculate the error
nm nano meter 1/1,000,000,000 of a meter or
10-9 m 10 Å
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Percent Error
(Measured Value - "Correct" Value)
Error
x 100
"Correct" Value
In the preceding example we measured a yellow
spectral line of Helium to be 12 cm and
calculated the corresponding wavelength to be
572.6 nm (5726.5 Å). Checking the spectrum for
He we find the chart wavelength, "correct", value
to be 587.6 nm.
(572.6 nm - 587.6 nm)
Error
x 100
587.6 nm
Error -2.6 the negative value tell us
that our calculated
value is lower than the actual value, a
positive
value would tell us that our calculated value is
higher
than the actual value.

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Newtons Form of
Keplers Third Law
(for the solar system)
(M m)
Where M m is the mass of the system, e.g. Sun
and planet
If one body is much more massive than the other,
M gtgt m, then
But theres a catch, the UNITS!
3
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Using Newtons Form of
Keplers Third Law
(for the solar system)
- to determine the mass of parent body, e.g.
Jupiter
3
3
a
R
But theres a catch when using this form, the
UNITS!
1st R or a, the semi-major axis must be AU and
2nd the Period must be given in Earth Years
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Calculate Mass of Star A with a planet -
orbiting at 6 AU or semi-major axis of 6 AU-
with a period of 3 Earth Years
A
B
1
Mass of Star A
6
Semi-major axis
2
3
a3
216
B23
Semi-major axis cubed
3
4
Period
5
9
B42
6
Mass
B3/B5
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Solar Masses
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Measuring the Height of a Mountain on the Moon or
using Ratios and Tangents,
Height
Angle
Length
If we knew the ratio of the height of the
mountain to the length of its shadow, then by
measuring the shadow (or counting pixels and
using the scale in our case) we could determine
the height of the mountain.
Height Length
Ratio
or Height Ratio x Length
Height Length
The ratio we are looking for is the
opposite side of a right triangle divided by
the adjacent side which is the tangent of the
angle shown. And tangents of angles are given in
trig tables. So Height
Tangent(angle) x Length of Shadow
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Density
D Mass/Volume gm/cm3
In the Meteorite Lab we will use the fact
that The density of water is 1 gm/cm3 to write
volume mass(in air) - mass(in water) D

mass(in air)
mass(in air) - mass(in water)
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Temperature Scales
Fahrenheit Celsius Kelvin
Boiling Point 212 100
373.15 Freezing Point 32
0 273.15 Absolute 0
-459.67 -273.15 0
9
5
F C 32 C (F-32)
K C 273.15
9
5
What is our normal body temperature in Celsius?
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TABLES AND GRAPHS
SI RABBIT POPULATION
160,000
140,000
120,000
100,000
RABBIT NUMBER
80,000
Series1
60,000
40,000
20,000
0
1600
1700
1800
1900
2000
2100
YEAR
DY RISE DX RUN
Y2-Y1
SLOPE
X2-X1