Don't you just hate it when newspapers write `a hundred thousand million miles to Arcturus' as if you were just going there to drop the kids off for soccer---Andy Rooney (apocryphal)
Sections
Frequently in introductory astronomy classes some students need a bit of a refresher in math and most students need introduction to some basics that are needed in astronomy.
This lecture is that refresher and introduction.
The math never gets any worse than this lecture---well almost never.
In astronomy very large and small number turn up all the time. Hence scientific notation is essential. Here is an example:
931=9.31 * 10**2 , where 10**2 means 102.
I got tired writing the html superscript tags and
they look bad in closely spaced lines.
So I use an old fortran notation of double asterisks **
to mean to be raised to.
Just accept it.
Some more examples of scientific notation:
1) c = 299879245800 cm/s is the vacuum speed of light which approximately = 2.998 * 10**10 cm/s = 2.998 * 10**8 m/s = 2.998 * 10**5 km/s . 2) amu = 0.000 000 000 000 000 000 000 000 001 6605 kg =1.6605 * 10**-27 kg is the Atomic Mass Unit (amu) which is about the mass of a hydrogen atom which is the lightest atom. 3) (9.31 * 10**2)*(2.998 * 10**10) =9.31 * 2.998 * 10**(2+10) =9.31 * 2.998 * 10**12 , and so exponents add on multiplication 4) (9.31 * 10**2)/(2.998 * 10**10) =(9.31/2.998) * 10**(2-10) =(9.31/2.998) * 10**-8 , and so exponents subtract on division
Note 3.00 * 10**10 implies that the prefix number is not 3.01 or 2.99.
If the number were more accurately known, it could be 3.004 or 2.996.
In this course, we do NOT worry much about significant figures or quantitative uncertainty estimates.
But they are essential at a higher level.
Using CONVENIENT UNITS is the usual rule.
In daily life mph, feet, Fahrenheit degrees are convenient enough---although they are not especially convenient: just reasonably so and, of course, traditional.
But for scientific and engineering purposes one needs units that are adapted to mathematical manipulation.
The main system today is the metric or SI system. There are two main subsets of metric units:
MKS = meters, kilograms, seconds:
used in most sciences and engineering
CGS = centimeters, grams, seconds:
used in astronomy---very backward of us.
I'll use either as suits my needs:
1 kg = 1000 g
1 m = 100 cm .
I often use kilometers too:
1 km = 1000 m = 10**5 cm .
There are other funny
metric prefixes to
pro/demote units by powers of ten. Some of them are rarely used.
``Mega'' symbolized by capital M means million or 10**6, for example.
Now MKS and CGS are basic reference systems of units that are good for calculation and comparisons of quantities that vary vastly in scale.
For special purposes one often uses units which are particularly suited to the physical system one is dealing with: i.e., one uses CONVENIENT UNITS.
Examples:
Thus the inch is the natural unit for dealing with the placement of items on a sheet of paper. Centimeters have always seemed pretty useless for dealing with paper sheets---they're too small.
This distance is called the astronomical unit or AU for short:
1 AU = 1.49597870*10**13 cm = about 1.496*10**13 cm = about 1.5*10**13 cm.
Answer 1 is right.
In this class, I ask you to memorize only a very few numbers---maybe only two or three or...
The first number to memorize is the Earth-Sun distance in AUs: it's ONE.
The Earth's equatorial radius is 6378.140 km: the polar radius is 6356.90 km---the Earth is a bit oblate.
In equatorial Earth radii, the mean distance of the Moon is
R_Moon= 60.2684 R_Earth_eq = 2.57*10**-3 AU .
60 Earth radii is the number I remember.
One can see that Earth and Moon are both pretty small compared to their separation distance.
The Earth-Moon system roughly to scale
(Cox-16,305).
The eccentricity of the Moon's elliptical orbit is 0.0549. See orbits below.]
Now there is one quantity whose astrophysically convenient measurement unit might be a bit obscure to you-all: that quantity is temperature.
We will never use the Fahrenheit scale in this class---except to comment on the weather outside.
The Celsius scale is probably familiar to you:
0 C is the freezing point of water (32 degrees Fahrenheit).
100 C is boiling point of water (212 degrees Fahrenheit).
(Actually 99.975 C at standard pressure which is 1 atm=1.013*10**5 Pa
[KWK-208].)
T_F=T_C*1.8 + 32 is the conversion from Celsius to Fahrenheit.
Sometimes we may use Celsius. But usually we'll use the Kelvin temperature scale or, as it is sometimes called, the ABSOLUTE temperature scale.
Kelvin degrees (with symbol K) are the same size as Celsius degrees, but the zero point of the Kelvin scale is absolute zero.
Temperature is in a sense a measure of random microscopic motion: i.e., atoms or molecules moving about in gases or liquids, or vibrating in solids; also random fields of electromagnetic energy (i.e., light).
If that microscopic motion reaches an irremovable minimum (called the zero-point energy in quantum mechanics), then you can't make make any less motion.
You've reached an absolute fundamental lower bound on microscopic motion.
We call that condition absolute zero.
Now for a macroscopic sample reaching absolute zero seems impossible, but microscopic samples can reach it.
And even without reaching it, you can find out easily enough where it is by a various limiting procedures---which are easy enough to do, but we won't discuss them here.
So absolute zero is in fact well known.
It is -273.15 C, in fact, or, as aforesaid 0 K (HRW-429).
T_K=T_C + 273.15
and
T_C=T_K - 273.15 .
Table of Notable Temperatures
Notable Temperature Kelvin (K) Celsius (C) Fahrenheit (F)
absolute zero 0 -273.15 not worth knowing coincidence 233.15 -40 -40 water freezing 273.15 0 32 human warmish 300 26.85 80.33 water boiling 373.125 99.975 212 pure iron melting 1808 1535 who cares pure iron boiling 3023 2750 " Sun surface 5800 who cares " Sun center 15*10**6 " "
For more details on temperature and the Kelvin scale see an Exposition on the Kelvin scale. Those details though are beyond the scope of this course.
In my view, we should just drop Fahrenheit and Celsius altogether and just use Kelvin all the time. This is my crank idea.
Rubaiyat of of Omar Khayyam, 5th edition, Verse XXXI by Omar Khayyam & Edward Fitzgerald
So we need to do a bit of math to gain some insight into the mathematical nature of astronomy.
Just addition, subtraction, multiplication, division, taking a square root, and a little algebra.
Typical kind of MATH we'll encounter is the calculation of speeds or times.
Let us consider some examples.
Redundantly, we can repeat the calculation.
v=(2 pi r)/(1 year)
= (2 * pi * 1.5 * 108 km)/(pi * 107 s)
=30 km/s ,
where
the circumference of a circle is 2*pi*r ,
r=1.5*108 km is the AU, of course ,
and
1 year= pi * 107 s to within .5 % .
This last result is just
a coincidence: there is nothing deep in it, but it is easy to
remember.
A more exact calculation of the Earth's mean orbital speed gives 29.785 km/s.
For comparison the escape speed from the Earth is 11.2 km/s (HRW-305).
The orbital speed of the Earth is determined by the motion of the Earth under gravity, but oddly enough it is almost independent of the Earth's mass (Go3-102).
But it DOES depend on the Sun's mass as we will discuss in IAWL: Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.
Answer 1 is right.
Since the only thing that distinguishes the Earth from other point-like masses as a gravitating body is its mass and the speed is almost independent of that, it follows that all bodies orbiting in the vicinity of the Earth with somewhat circular orbits will be moving at about 30 km/s.
So of order 30 km/s is about the speed of any asteroid that would hit us.
Answer 2 is right.
An impactor coming from more or less behind would have a relatively low relative speed; one coming head on would have a high relative speed approaching of order 60 km/s.
BEHOLD
d=vt, and so t=d/v=1.496*10**13 cm / 3*10**10 cm/s
=500 s=8 min, 20 sec .
So about 8 minutes.
If the Sun blew up right now, we'd live in blissful ignorance for about 8 minutes.
This problem is one of a general class where you have an amount A and a rate of change R and are asked how long till the amount A is used up.
BEHOLD
A=Rt, and so t=A/R .
If those numbers are treated as hard, then there are about 1000/30=33 years before all the conventional oil in the world is gone.
Of course, things are NOT that simple.
Maybe we will not burn the last drop of oil, but move to a hydrogen and renewables economy.
Still conventional oil might become a lot more expensive as the next decades role by.
See for example the End of Cheap Oil.
d=ct=(3.00*10**10 cm/s) * (pi*10**7 s)
=9.4*10**17 cm = about 10**18 cm ,
or more exactly 9.46053*10**17 cm.
This, of course, is one light-year (lyr).
Note astronomers (for reasons I am not prepared to discuss) often use parsecs rather than light-years.
1 parsec=3.0856776*10**18 cm
=3.26 lyr = about 3 lyr
(Cox-12).
amount
Amount = Rate * time and time = ________
rate
Special case examples of these are in calculating distance d
traveled at speed v in time t and travel time t at speed v over
distance d:
d
d = v * t and t = ___
v
From the Babylonians circa 500 BC (No-39), the circle is divided into 360 equal bits: i.e., 360 degrees.
The Babylonians did NOT say why, but some speculations exist: see Mesopotamia and Angular Measurement.
Alas, the French Revolution that gave us the metric system completely overlooked angular measure, and so we're stuck with 360 degrees.
There are some finer units that we use occasionally:
1 degree = 60 arcminutes
= 3600 arcseconds
and
1 arcminute = 60 arcseconds .
Just for general astronomical interest, one can make simple angle
measurements with your hands.
Angle measurements with a hand.
Now everyone's hand is a bit different---we are all unique---but just approximately at arm's length
1 finger = 1 degree ,
1 fist = 10 degrees , and
1 spread hand = 18 degrees
(Se-18).
These results convenient for judging angles on the sky. Examples of astronomical angles are:
An angular diameter is the angle subtended by the diameter of a spherical body?
Answer 3 is right.
The Sun and the Moon have very different sizes: the Sun diameter is about 400 times the Moon's diameter, but the Sun is about 400 times further away than the Moon.
This is the GREAT COINCIDENCE which seemed for long ages to have great cosmic significance, but is now seen just as an accident of solar system formation and evolution.
The upshot is the Sun and the Moon have almost the same angular diameter on the sky: i.e., about 0.5 degrees.
More exactly as seen from the Earth's center using mean distances to Sun and Moon and mean diameters we find:
di_moon=0.515127 degrees
and
di_sun=0.533121 degrees.
Answer 1 is right.
There is a sky coordinate system analogous to longitude and latitude that we will very briefly discuss in IAWL Lecture 2: The Sky.
Even the Ancients could measure angles fairly accurately---when they weren't being sloppy that is---and today accuracy to better than 1 arcsecond (i.e., subarcsecond accuracy) is pretty common.
But the sky has NO APPARENT DEPTH, except that it's far.
There is no simple way to tell distances by eye or even by simple geometric ways available to the Ancients.
Even today distance measurements are relatively hard---relative to angular measurements.
If you have angular position, you can have ANGULAR VELOCITY.
For example say that the angular speed is a constant, then the angular speed just equals any change in angle divided by the corresponding change in time:
Delta theta
____________ = angular velocity ,
Delta time
where Delta means change in.
Physics and astronomy often use Greek letters to represent standard items
But html doesn't support Greek letters.
So, except in diagrams, I must spell the letters rather than write them.
For reference, the complete Greek alphabet---the alpha to the omega---is presented below with Delta in 4th place.
The Greek Alphabet: alpha, beta, gamma, ...
Alpha, beta, ... --- do you get it?
Or from Earth's perspective, what is the angular speed of the Sun around the Earth measuring the Sun relative to the ``fixed stars''?
Behold
Delta theta 360 degrees
____________ = ______________ = about 1 degree/day .
Delta time 365.25 days
Actually it's a little less than 1 degree per day.
For some speculation see Mesopotamia and Angular Measurement.]
We often have to show plots and graphs---which are synonyms---in this course.
It's good to qualitatively recognize certain function behaviors on graphs.
Some simple function behaviors.
We also sometimes encounter logarithmic plots or log plots.
You do NOT have to know what a logarithm is to appreciate log plots.
On a log plot the unit is some power of 10: e.g., 10**(1/2), 10, 10**2, 10**3, etc.
Log plots are useful for showing quantities that vary over many orders of magnitude.
On ordinary plots much of the quantity's behavior is off the plot or is squashed down to the horizontal axis.
Just a first discussion now.
We'll look at motion and orbits again when we do a deeper discussion of Newtonian physics and gravity in IAWL: Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.
From a geometrical point of view, any position can be taken to define the origin of a motionless reference frame.
A reference frame just be a set of coordinates covering all space.
But there are physically preferred frames.
For example, nowadays were almost always say the Earth orbits the Sun even though geometrically the reverse is equally true and we DO sometimes take the GEOCENTRIC perspective.
The Sun's reference frame is physically preferred.
This is because Sun's reference frame is a closer approximation to being an inertial frame than the Earth's frame of reference.
Inertial frames are fundamentally unaccelerated frames.
In modern physics, an acceleration is a change in speed and/or a change in direction.
And it is acceleration which is physically noticeable.
Rest and constant-speed straight-line motion (i.e., unaccelerated motion) are not physical distinct: which is which depends on your inertial frame of reference.
The causes of accelerations are forces.
In fact the definition of force is that it is a physical relationship between bodies that can cause an acceleration.
Because it is accelerated, a rotating frame cannot be exactly an inertial frame.
Newton's laws are defined relative to inertial frame.
It is hard to find an absolutely inertial frame:
Actually, orbital period of the Sun about the center of the Milky Way. is about 220 million years or 220 Myr (FK-565).
The fixed stars are NOT really fixed: ``fixed stars'' is a historical term to distinguish relatively unmoving stars from moving planets.
Newton posited the fixed stars as being the exactly inertial: we've had to change that assumption since his day.
We can, in fact, identify our motion with respect to such a frame using cosmic microwave background (CMB) (see also IAWL Lecture 31: Cosmology) and quasars.
In the more inertial Sun frame it is the planets that orbit the Sun.
Why is this frame is more inertial?
As we will discuss in IAWL Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides, acceleration is proportional to force and INVERSELY proportional to mass.
The gravitational forces between Sun and planets are equal and opposite by Newton's 3rd law.
But the Sun dominates the mass of the solar system.
The whole planetary system has mass of only 0.134 % of the Sun's mass (Cox-293).
So the Sun is much LESS accelerated than the planets and the Sun frame is much more inertial than their frames.
This has been a long story, but it explains why we say ``the planets orbit the Sun.''
If the Sun suddenly disappeared, the planets would fly away from each other in space and never meet again because the major source of gravity was gone: gravity is proportional to mass.
The Sun dominates the solar system motions.
What are the orbital shapes?
Well planet orbits are NEARLY CIRCLES about the Sun and moon orbits are NEARLY CIRCLES about their respective planets in most cases.
Actually, in finer detail, planet, moon, and asteroid orbits are all approximately ellipses.
There's a mathematical formula for ellipses, of course---but I don't what to shock and awe the students.
Instead let's just consider ellipse eccentricity
Eccentricity is a measure of the stretched-outedness of an ellipse.
Again there is a formula connecting eccentricity and ellipse shape, but experience suggests students don't want to know it.
How do these mathematical ellipses connect up with orbits?
Say you have a TWO-BODY GRAVITATING SYSTEM in which one body is MUCH more massive that than the other.
In such a TWO-BODY SYSTEM, the less massive body orbits the more massive body in an ellipse with the the more massive body at one focus.
The other focus is just an empty point in space.
Newtonian physics make this happen.
We'll look at Newtonian physics later in more detail in IAWL Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides, but we won't derive this ELLIPTICAL ORBIT RESULT result which is actually non-trivial---it gave Newton a hard time.
The center of mass is the mass-weighted average position of the two bodies. We won't give a formula here.
But if one body is much more massive, it effectively is the center of mass and is at the focus of the other bodies orbit.
This is the case we've just discussed.]
The eccentricity e turns out to the relative variation of the perihelion and aphelion distances from the mean distance.
For example, if e=0.2, then the perihelion distance is 20 % smaller and the aphelion distance is 20 % larger than the mean distance.
Planets move fastest at perihelion and slowest at aphelion.
The names perihelion and aphelion apply strictly only to orbits about the Sun.
But same special points occur in any elliptical orbit.
For orbits about Earth, one uses perigee and apogee: GEE is derived from the Greek for Earth.
For orbits about a star, one uses periastron and apastron: astron just being derived from Latin for star.
Probably, one could just use perigee and apogee or periastron and apastron generally.
Sometimes people use very specific names for the points: e.g., PERI-JUPITER and AP-JUPITER for orbits about Jupiter.
Similarly, the planets all pretty much dominate their moon systems, and so a planet and each of its moons is approximately a TWO-BODY SYSTEM that are analogous to the Sun-planet systems.
Now, of course, to consider Sun and each planet or planet and each moon as TWO-BODY SYSTEMS is an approximation.
All bodies in the solar system interact by gravity with all others.
Every bit of mass gravitationally attracts every other bit: this attraction is weak for small-mass bodies, but large for large-mass ones. It falls off by an inverse-square law with distance: F proportional to 1/r**2, where r is the distance separating the body centers.
This makes the real interaction immensely complex.
---Isaac Newton circa 1684 in De Motu, quoted from Cohen & Whitman's Newton's Principia (CW-18).
Well maybe the calculation isn't easy, but it can be done.
You first solve for the dominant effect on each body using the TWO-BODY SYSTEM approximation.
Then you correct your calculation, by adding on the effects of the other bodies in order of decreasing importance.
Thus, by a series of corrections you approximate the true behavior of the planets and moons more and more closely.
The secondary effects are called perturbations.
Solving a multi-body system using perturbations.
In this course we will often just say ``perturbations do it'' to explain fine details.
The perturbations of the secondary gravitational sources on bodies cause all the orbits to be NOT exactly ellipses and NOT exactly constant in time.
The solar system motions are not a perfect repeating clockwork although over the course of human history they approximate that.
The solar system is in fact slowly evolving.
The evolution is actually chaotic: i.e., this means super-sensitive to initial conditions among other things.
But for millions or even billions of years the changes in the orbits of the solar system are small and in particular the major bodies of solar system are approximately predictable like clocks if you have a sufficiently sophisticated computer program.
The smaller the body the less predictable in general, because smaller bodies are more easily affected by the many weak effects (e.g., gravity perturbations and light pressure from the Sun).
More mathematical details about ellipses and elliptical orbits are given on an ellipse page.
Those details are NOT required for this course.