This is power point 2

1. Math
for
Astrophysics
Lecture
2
AST
102,
Prof.
Rice
Wednesday
10:10-­‐11:00
am
September
11,
2013

2. By
the
end
of
this
class
you
will
be
able
to…
• Explain
why
scienIfic
notaIon
is
necessary
in
astronomical
contexts
• Translate
astronomical
numbers
between
scienIfic
notaIon
and
wriKen
English
• Do
math
with
numbers
in
scienIfic
notaIon
• Work
with
metric
system
units
of
measure
• Manipulate
raIos
and
proporIonaliIes

3. Numbers
in
astrophysics
are
astronomical…
• 200,000,000,000
stars
in
the
Milky
Way
Galaxy
• 100,000,000,000
galaxies
in
the
Universe
• 10,000,000,000,000,000,000,000
stars
in
the
Universe
• 100,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000
atoms
in
the
Universe
Other
units
in
astrophysics
span
many
orders
of
magnitude,
too

4. OpIon
1:
ScienIfic
NotaIon
• WriKen
as:
Y
×
10X
• Y
is
a
number
between
1
and
10
• X
is
the
exponent,
or
power
• Can
be
a
posiIve
or
negaIve
number
• X
>
0
à
zeros
to
the
leZ
of
the
decimal
point
(
>
1)
• X
<
0
à
zeros
to
the
right
of
the
decimal
point
(
<
1)
• X
=
0
à
100
=
1

5. Math
with
Exponents
is
slightly
different…
MulIply
à
add
the
exponents
104
×
107
=
10(4+7)
=
1011
Divide
à
subtract
the
boKom
exponent
104
/107
=
104
×
10-­‐7
=
10(4-­‐7)
=
10-­‐3

6. (Review
mulIplicaIon
&
fracIons
if
you
need
to)
×,
x,
*,
()()
or,
e.g.
3π
=
MulIply
BoKom
of
the
fracIon
is
a
negaIve
exponent
“BoKom
of
the
boKom”
of
a
fracIon
à
top

7. An
Order
of
Magnitude
is
a
factor
of
10
Power
Value
Name
Power
Value
Name
101
10
Ten
10-­‐1
0.1
Tenth
102
100
Hundred
10-­‐2
0.01
Hundredth
103
1,000
Thousand
10-­‐3
0.001
Thousandth
106
1,000,000
Million
10-­‐6
0.000001
Millionth
109
1,000,000,000
Billion
10-­‐9
0.000000001
Billionth
1012
1,000,000,000,000
Trillion
10-­‐12
0.000000000001
Trillionth
A
nega3ve
exponent
is
equivalent
to
the
same
number
with
a
posiIve
exponent
on
the
boKom
of
a
fracIon.
For
example:
10-­‐3
=
1
/
103
=
1
/
1,000
=
0.001,
or
“one
thousandth”

8. The
Metric
System
• Basic
units
in
the
metric
system
are
meters,
kilograms,
and
seconds
• Prefixes
represent
powers
of
ten
in
these
units
You
will
need
to
know:
CenI-­‐
10-­‐2
Milli-­‐
10-­‐3
Kilo-­‐
103
Micro-­‐
10-­‐6
Mega-­‐
106
Nano-­‐
10-­‐9
Giga-­‐
109

9. OpIon
2:
Change
Units
• Distances
within
a
stellar
system
are
given
in
AU
AU
=
Astronomical
Unit
=
average
distance
between
Earth
&
Sun
1
AU
=
92,955,887.6
miles
“one
Astronomical
Unit
is
almost
93
million
miles”
• Distances
between
stars
are
given
in
light
years
1
light
year
=
5,878,625,373,183.608
miles
“one
light
year
is
almost
six
trillion
miles”
• Mass
is
expressed
in
terms
of
a
well-­‐known
object:
1
MSun
=
1.99
×
1030
kilograms
1
MJupiter
=
1.90
×
1027
kilograms
1
MEarth
=
5.97
×
1024
kilograms

10. Changing
units
requires
a
conversion
factor
• A
conversion
factor
is
a
fracIon
equal
to
one
with
different
units
on
the
top
and
boKom
– Say
a
movie
is
120
minutes
long
…
how
many
hours
is
that?
– 1
hour
=
60
minutes
becomes
a
conversion
factor
• How
far
from
the
Sun
is
Mars
in
AU?
– Average
distance
is
142
million
miles
– 1
AU
=
92,955,887.6
miles
Distance
=
1.42
×
108
miles
×
1
AU
=
(1.42/9.3)
×
101
AU
9.3
×
107
miles
≈
1.5
AU

11. How
many
miles
are
in
1
AU,
expressed
in
scienIfic
notaIon?
9.3
×
107
93
×
106
93,000,000
0.93
×
108

12. How
many
miles
are
in
1
AU,
expressed
in
scienIfic
notaIon?
9.3
×
107
Step
1:
93
million
miles
à
93,000,000
Step
2:
93,000,000
=
9.3
×
107

13. How
far
is
Neptune
from
the
Sun
in
miles?
• First,
handle
coefficients
• Then,
handle
exponents
• Last,
combine
coefficient
and
exponent
Average
distance
of
Neptune
is
30
AU
à
3.0
×
101
AU
1
AU
=
93
million
miles
à
9.3
×
107
miles
Step
1:
3.0
×
9.3
=
27.9
Step
2:
101
×
107
=
10(1+7)=108
Step
3:
27.9
×
108
=
2.79
×
109
Step
4:
2.79
×
109
à
3
billion
miles

14. We
can
use
ra3os
to
make
large
numbers
more
understandable
Example:
How
many
Imes
more
massive
is
the
Sun
than
Jupiter?
1
MSun
=
1.99
×
1030
kg
1
MJupiter
=
1.90
×
1027
kg
The
Sun
is
about
1,000
Imes
more
massive
than
Jupiter.

15. How
many
Imes
more
massive
is
the
Sun
than
Earth?
1
MSun
=
1.99
×
1030
kg
1
MEarth
=
5.97
×
1024
kg
one
hundred
million
one
hundred
thousandth
three
hundred
thousand
two
million

16. How
many
Imes
more
massive
is
the
Sun
than
Earth?
1
MSun
=
1.99
×
1030
kg
1
MEarth
=
5.97
×
1024
kg
three
hundred
thousand
Step
1:
1.99/5.97
≈
2/6
=
0.333
Step
2:
1030/1024
=
10(30-­‐24)=106
Step
3:
0.333
×
106
=
3.33
×
105
Step
4:
3.33
×
105
à
330,000

17. It
takes
about
two
hours
to
drive
from
CSI
to
AtlanIc
City.
How
long
will
it
take
if
you
drive
twice
as
fast?
Time
=
Distance
/
Velocity
Ta
=
Da
/
Va
What
stays
the
same?
Ta
∝
1
/
Va
Vb
=
2
Va
Tb
∝
1
/
Vb
=
1
/2Va
=
½
Ta
We
can
use
propor3onali3es
to
make
equaIons
more
understandable

18. The
volume
of
a
sphere
is
V
=
4/3
π
R3.
Which
proporIonality
is
correct?
V
∝
π
R3
V
∝
4/3
π
R3
V
∝
4/3
R3
V
∝
R3

19. Sphere
A
has
a
radius
twice
as
large
as
Sphere
B.
What
is
the
raIo
of
volumes
of
the
spheres?
Va
/
Vb
=
1/8
Va
/
Vb
=
8
Va
/
Vb
=
4/3
Va
/
Vb
=
3/4

20. For
Next
Class
• Read
SecIons
4.1-­‐2
for
How
The
Universe
Works:
MoIon
&
Energy
• Mastering
Astronomy
Homework
1
due
Sunday,
September
15
• Register
for
Mastering
Astronomy,
take
HW
poll,
sign
up
for
Dropbox