Math Tools
O. Pretest Answers:
-- seasons are caused by the tilt of the Earth's axis relative to its
orbit, and the fact that the tilt makes the Sun higher in the sky
and stay up longer in Summer. They are not caused on Earth
by anything that has to do with distance to the Sun, in fact you
are closer to the Sun right now than any other time.
-- the phases of the Moon are caused by the relation between the Moon
and the Sun, relative to the Earth. This relation changes throughout
a 28.5 day cycle due to the Moon's orbit around Earth. Phases are
not caused by anything to do with the shadow of the Earth,
they are in fact caused by the Moon's own shadow.
-- Earth satellites do not fall on our heads, although they are indeed falling,
because they move sideways fast enough to continually "miss" the horizon.
This is called an orbit. It is not gravity that keeps them from falling,
gravity keeps them from flying off into space. The most common wrong
answer is that there is no gravity in space, and a lot of this course will be
devoted to talking about the fact that there is plenty of gravity in
space, and in fact it is the most important force in astronomy.
I. Scales in Astronomy
-- concept of a scale model, and the scale model of the
solar system
-- note the key characteristic of a scale model is that it
gives the same result as the real thing for all ratios
-- to handle the huge differences in scale in astronomy, we often
count by powers of 10, which relates to the number of digits in
the number represented in the decimal system. So we don't count
1, 2, 3, astronomers count 10, 100, 1000, etc.
II. Large numbers and "scientific notation"
III. Units
-- importance of quoting units with every number
-- how to convert units: "multiplying by ones"
IV. Meaning of =
-- importance of equations in science
-- versatility of =
* is identical to
* is defined as
* is equivalent to
* is conditionally equal to
V. Manipulating equations
-- start with true equations you can "take to the bank"
-- do the same thing to both sides
-- make substitutions
VI. Meaning of a formula
-- constants vs. variables
-- understanding the implied dependences (scalings)
-- recipe for solving dependence problems
-- a formula is just an ultracompact way to talk about how
things depend on other things, and most formulas connect
with essential concepts