--- The key point to this whole lecture.
Sections
This lecture is that refresher.
Note the courses supported by IAL are NOT math intensive, but astronomy is, and so to get some flavor of that some math needed in IAL. Also any science should refresh/stretch the math skills of students a bit.
The math never gets any worse than in IAL 1---well maybe never---don't want to be too categorical.
You all should think of yours truly as Euclid (fl. 300 BCE) in the figure below (local link / general link: euclid.html).
Actually, the googolplex is a bit large even in astronomy.
That way is scientific notation---which we have already just used above for the googolplex.
In scientific notation, a number is written in the form
Note fortran is NOT ...
See the figure below
(local link /
general link: alien_fortran_short.html).
Here are some examples
of scientific notation:
In modern physics and metrology, the vacuum light speed is an exact value by definition.
Memorable approximations for the vacuum light speed:
= 2.998 * 10**8 m/s
= 2.998 * 10**5 km/s
≅ 3 * 10**5 km/s
≅ 1 ft/ns (which is often well known to people
building precision circuitry and optical systems).
We illustrate the vacuum light speed in the figure below (local link / general link: light_speed_earth_moon.html).
The AMU is defined as exactly
(1/12) of the mass of
the unperturbed carbon-12 (C-12) atom---all
unperturbed C-12 atoms are exactly
alike IN PRINCIPLE according to
quantum mechanics.
Now an unperturbed atom
is an ideal limit that CANNOT be exactly reached in practice, but it can be
very, very closely approached easily which makes the
definition of the AMU very, very
useful.
The AMU is also
approximately the mass of
the hydrogen atom which is
the lightest atom.
By the way, atomic hydrogen gas
(i.e., a gas
of hydrogen atoms,
NOT a gas
of hydrogen molecules
H_2) can be identified by
its emission line spectrum
using spectroscopy.
As a preview, the figure below
(local link /
general link: line_spectrum_hydrogen_balmer.html)
gives
the atomic hydrogen gas
emission line spectrum in the
visible band (fiducial range 0.4--0.7 μm).
We take up the subject of spectroscopy
in IAL 7: Spectra.
If the number were more accurately known, it could be 3.004 or 2.996.
In IAL,
we do NOT worry much about
significant figures
or quantitative uncertainty estimates.
But they are essential
at a higher level.
= [1.660 539 066 60(50)]*10**(-27) kg
≅ [1.660 5]*10**(-27) kg
is the which is the
atomic mass unit (AMU, Dalton, Da) = (1/12) C-12
= [1.660 539 066 60(50)]*10**(-27) kg
(see NIST:
Fundamental Physical Constants --- Complete Listing).
Actually, all microscopic scale
particles of a given type have
NO identity.
They CANNOT be told apart IN PRINCIPLE.
Quantum mechanics
dictates this.
Microscopic scale
particles have NO age either.
One a gigayear
old and one a second old, are exactly identical.
If you confused them spatially, you CANNOT tell which was old and which was young.
So age has no meaning for them.
Why the carbon-12 (C-12) atom and NOT
some other unperturbed atom since
all unperturbed atom of a given species
are exactly
alike in principle according to
quantum mechanics.
Oh, just convenience in practical measurement.
(a*10**b)*(c*10**d) = a*c*10**(b+d)
and so exponents add on multiplication
(a*10**b)/(c*10**d) = (a/c)*10**(b-d)
and so exponents subtract on division.
(9.31 * 10**2)*(2.998 * 10**10)
=9.31 * 2.998 * 10**(2+10)
=9.31 * 2.998 * 10**12 .
(9.31 * 10**2)/(2.998 * 10**10)
=(9.31/2.998) * 10**(2-10)
=(9.31/2.998) * 10**(-8) .
Using CONVENIENT units is the usual rule.
In everyday life, miles per hour (mph), feet, Fahrenheit degrees, etc. are convenient enough---although they are NOT especially convenient: just reasonably so and, of course, traditional in the US.
For everyday life, see the figure below (local link / general link: everydaylife_tv.html).
These units
(miles per hour (mph),
etc.), in fact, belong to the system of
United States customary units---which
we call British units since they are the
units
the British used to
use---the British---"One if by land, two if by sea" ...
In IAL, we will almost never use US customary units. They are NOT suitable at all for scientific purposes since it is hard to do scientific calculations with them and almost no one does anymore.
For scientific
and engineering purposes,
one wants units
that are adapted to mathematical manipulation by being
a decimal system based.
The main system today---and for the foreseeable future---for
scientific,
engineering,
and, outside of the US, civil purposes is the
Metric System (AKA SI).
I'll use either as suits my needs.
Here are some useful conversions:
There are funny
metric prefixes
that pro/demote
fiducial
metric units
by powers of ten: e.g.,
the prefix mega
symbolized by capital M promotes by a
million or 10**6 as illustrated in the figure below
(local link /
general link: alien_metric_mega.html).
Now metric system
is basic reference systems of units
that is good
for calculations and comparisons of
quantities that vary vastly in scale.
But 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
or, in science jargon,
natural units.
But it allows it to be used for convenient units by implication.
The natural unit
for any particular quantity
in a particular context is the amount of that quantity
possessed by a characteristic thing (ideally the most characteristic thing) in that context.
So natural units usually
highlight differences in amounts that are important and/or memorable, and so are useful
when thinking about quantities.
In fact, there is usually NO perfect
natural unit
for a quantity
and the choice of natural unit
is often based on humankind's perspective
and/or has a random element which is often historical or whimsical.
In the following subsections, we give
examples
of natural units.
The standard
North American letter paper size
is 11 X 8.5 inches. See the figure below
(local link /
general link: alien_natural_unit_inch.html).
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
sheets of
paper---they're too small.
To introduce the astronomical unit (AU),
see the figure below
(local link /
general link: astronomical_unit.html).
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 mean
Earth-Sun distance
in AUs: it's ONE.
As you can see from
Table: Solar-System Planets
below
(local link /
general link: table_solar_system_planets.html),
it is much easier to remember, comprehend, and contemplate
Solar System astronomical distances in
AU
than in kilometers, centimeters, or miles.
Note that eccentricity is
discussed below in section Orbits.
In dealing with the
Earth-Moon system
and general near-Earth
astronomical objects (natural or artificial),
it is sometimes
convenient to know the distances in the
natural unit of the
Earth equatorial radius R_eq_⊕ = 6378.1370 km.
The figure below
(local link /
general link: earth_oblate_spheroid.html)
explicates the Earth radii.
The natural units
for stars (see the figure below:
local link /
general link: night_sky_california_piper_mountain.html)
are set by the Sun: these are the
solar units.
The Earth unit (EU)
is explicated in the figure below
(local link /
general link: map_world_physical_EU.html):
"It's all my own invention."
Actually, American engineering uses a fiendish mixture of
metric units AND
United States customary units.
This has led to some blunders.
In fact, most of the world uses the
Metric System for most purposes.
See the figure below
(local link /
general link: metric_world.html).
There are two main subsets of
metric units:
MKS units
= meters, kilograms, seconds:
used in most sciences and engineering
CGS units
= centimeters, grams, seconds:
used in astronomy---very backward of us.
1 kg = 1000 g
1 m = 100 cm .
I often use kilometers too:
1 km = 1000 m = 10**5 cm .
We give the metric prefixes
in Table: Metric Prefixes below
(local link /
general link: metric_prefix.html).
By the by, whenever we look at
tables, the point is NOT to try to memorize them, but to contemplate what they mean
while looking at them.
Some of the
metric prefixes
are used rarely and maybe a few NOT at all.
Actually, Wikipedia---the
supreme authority---is formally more
restrictive about the use of the term
"natural unit".
Natural units
are NOT usually good for calculations.
For those, one usually
needs standard units: i.e., the metric system units.
Natural units
are good for thinking purposes and plotting purposes.
Question: What is the mean
Earth-Sun
distance in astronomical units?
Answer 1 is right.
Also, of course, it is easier to understand a
Solar System image too
when thinking in terms of
astronomical units (AU).
For example, see the figure below
(local link /
general link: solar_system_inner.html).
The figure below
(local link /
general link: earth_moon_system.html)
gives the
Earth-Moon
distance in
Earth equatorial radii
and compares that distance to the
astronomical unit.
Solar units
are explicated in the figure below
(local link /
general link: star_natural_units_solar_units.html).
That natural system of units is the Kelvin scale whose unit the kelvin (K) is somewhat natural because has the same size as the Celsius degree (C) which is a natural unit for biota.
We will now elucidate Kelvin scale.
We will NEVER use the Fahrenheit scale in this class---except to comment on the weather outside---e.g., it'll max at 110 F today (e.g., 2022 Sep06: see Weather Las Vegas)---but that's nothing to us in Las Vegas---or maybe 120° F (see Las Vegas temperature record: Historic Heat Wave in Las Vegas---Breaking Down the 120° F Record with the National Weather Service | 4:57).
0 C is the freezing point of water (32 degrees Fahrenheit).
100 C is boiling point of water (212 degrees Fahrenheit).
These are fiducial values
for typical Earth's atmosphere
pressure.
Exact values vary with pressure
and purity of water.
Note: standard atmosphere pressure = 1 atm = 101.325 kPa = 1.01325 bar = 14.696 Psi
which is also the
mean sea-level pressure.
T_F = T_C*1.8 + 32 is the conversion from Celsius to Fahrenheit.
It is the natural temperature scale for life as we know it (i.e., biota as we know it) since life as we know it requires liquid water to exist. Liquid water must be is possible at least inside the body.
It's also the natural temperature scale for humankind that prime example of life as we know it.
It is really simple to understand after Celsius scale since the kelvin degree (K) is the same size Celsius degree (C). The difference is in the zero point which for the Kelvin scale is absolute zero.
This makes the Kelvin scale the natural temperature scale for most of the purposes of physics and astrophysics.
For the eponym of the Kelvin scale, see the figure below (local link / general link: lord_kelvin.html).
Temperature
is, among other things, a measure of random
microscopic motion:
i.e., atoms or molecules moving about in gases or liquids,
or vibrating in solids.
Or in physics jargon, a measure of
kinetic energy
(the energy of motion).
If the microscopic motion reaches an irremovable minimum (called the
zero-point energy in
quantum mechanics),
then you CANNOT make make any less motion.
You've reached an absolute fundamental lower bound on microscopic motion.
See the figure below
(local link /
general link: temperature_microscopic.html).
We call that absolute fundamental lower bound on microscopic motion
absolute zero.
So cold, colder, coldest = absolute zero.
So can absolute zero be reached?
Yes and no.
It's a matter of point of view.
So absolute zero
is in fact well known.
It is -273.15 C, in fact, or, as aforesaid, 0 K
(absolute zero).
However, there are
negative temperatures
on the Kelvin scale.
This remarkable fact is explicated in the figure below
(local link /
general link: 1919_solar_eclipse_negative_thermo.html).
The
conversions
of the 3 standard
temperature scale
are given in the figure below
(local link /
general link: alien_kelvin.html).
Note that in physics jargon,
microscopic
means the about the size of atoms
or molecules or smaller.
The term is used loosely.
The animation in the figure below
(local link /
general link: gas_animation.html)
illustrates gas
molecules with
kinetic energy
and temperature above
absolute zero.
Although it is NOT obvious how, microscopic motion
atoms
and
molecules
gives us our sense of
hot and cold.
The science of perception of physical stimuli is
psychophysics---which is its own vast realm.
Now for a macroscopic sample reaching
absolute zero seems
impossible, but small enough microscopic samples
can reach it.
Some say microscopic
samples CANNOT reach it either since
temperature ceases
to be a well-defined quantity if the sample gets too small.
Temperature
in their view
is a macroscopic
measure of average macroscopic behavior.
However, without reaching absolute zero,
you can find out easily enough where
it is by a various limiting procedures---which are easy enough
to do, but we will NOT discuss them here.
Below in
Table: Temperature Scale Comparison for Notable Temperature States
(local link /
general link: table_temperature_scale_comparison.html), we compare
the Kelvin scale,
the Celsius scale,
and the Fahrenheit scale for notable
temperature states.
Form groups of 2 or 3---NOT more---and tackle Homework 1 problems 7--16 on units and natural units.
Discuss each problem and come to a group answer.
Let's work for 5 or so minutes.
The winners get chocolates.
See Solutions 1.
Rubaiyat of Omar Khayyam, 5th edition, Verse LVII by Omar Khayyam (1048--1123) and Edward FitzGerald (1809--1883).
So we need to do a bit of math to gain some insight into the mathematical nature of astronomy. Just few tools: see the figure below (local link / general link: franklin_d_roosevelt.html).
Just
addition,
subtraction,
multiplication,
division,
taking a square root,
a little algebra,
and geometry, and ...
So no need for
fear
and loathing.
The typical kind of math
we'll encounter is the calculation of speeds or times.
Let us consider some examples.
What is the speed of the Earth around the Sun in the inertial frame of the Solar System (i.e., the celestial frame of the Solar System) in kilometers/second (km/s). The answer is illustrated in the figure below (local link / general link: earth_orbital_speed.html).
Why use km/s? They are the natural unit people use for Solar System and other astrophysical velocities as we will discuss below.
Redundantly with the figure above
(local link /
general link: earth_orbital_speed.html),
we repeat the calculation of
Earth oribital speed below.
A speed is ratio: distance over time: thus
v = (2πr)/(1 year)
= (2 * π * 1.5 * 108 km)/(π * 107 s)
= 30 km/s ,
where
the circumference of a circle is 2πr ,
r = 1.5*10**8 km is the astronomical unit, of course,
and
1 year = π * 10**7 s to within 0.5 % .
That 1 year ≅ π * 10**7 s 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 29.783 km/s (Wikipedia: Earth).
The kilometer per second (km/s) is, in fact, a natural unit for orbital velocities and many other macroscopic velocities in the astrophycial realm in the general.
The kilometer per second (km/s) is a natural unit since has a convenient size for thinking about these macroscopic velocities:
For galaxies, orbital velocities are typically of order hundreds of kilometers per second.
The ejected material in supernova explosions is typically of order thousands to tens of thousands of kilometers per second.
Thousands and tens of thousands are getting to be biggish numbers, but they still trip off the tongue unlike 10**9 cm/s.
Of course, the kilometer per second (km/s) is NOT the natural unit in all cases---but pretty often.
But oddly enough, the orbital speed is ALMOST INDEPENDENT the Earth's mass since that mass is much smaller than that of Sun which it does depend on. We discuss this point further below in section Orbits and in IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides (see also Go3-102).
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.
We might sometimes want to calculate
the circular orbit velocity
and escape velocity
(i.e., escape orbit velocity) in general.
In any case, it's worth having a look at the general
formulae for these quantities
and also the natural unit,
the kilometer per second,
for many astrophysical systems.
We will never ask students to memorize these formulae,
but it is useful to see what they look like since they are relatively simple rather than just say they
exist without any idea of their appearance.
For the formulae,
see the insert below
(local link /
general link: orbit_velocity_circular_escape.html).
Note from the formulae above that the ideal
low Earth orbit
circular orbit velocity
v_circular = 7.9053 km/s
and
the ideal Earth
escape velocity
v_escape = 11.180 km/s
are really, really fast.
At t = 0, you are here and at t = 1 s, you are 7.9 or 11.2 km away depending which case
you are considering.
In everyday life, we do NOT
encounter such velocities.
The very high
circular orbit velocity
is why it's hard to get to low Earth orbit.
It takes large rockets to get to ∼ 7.9 km/s.
In fact, actual
lowest low Earth orbits
at altitude ∼ 200 km have
circular orbit velocities
∼ 7.8 km/s (somehow a bit less than ∼ 7.9 km/s maybe due to the non-uniform
gravitational field
of the Earth)
and require launch velocities
of ∼ 9.4 km/s
to overcome
air drag
(of the Earth's atmosphere),
loss of kinetic energy to
gravitational potential energy
in raising the spacecraft
to altitude ∼ 200 km,
and probably other complications
(see Wikipedia: Low Earth Orbit:
Orbital characteristics).
The required speeds are much too high for
jet aircraft---even if they
could dispense with having air---which
they can't, unless they become semi-rockets.
The fastest true jet, the
Lockheed SR-71 Blackbird
(now retired: see
Wikipedia; Lockheed SR-71 Blackbird:
Final retirement),
reached only Mach 3.3.
Mach 3.3 ≅ 1.1 km/s.
Note that
Mach number is speed in units of
local sound speed, and thus
Mach number is the
natural unit
of flight.
For flight,
see the figure below
(local link /
general link: flight_wright_flyer.html).
Note also that
The sound speed is 343.2 m/s
at 20°C in dry air.
Sound speed varies significantly
with temperature and
humidity.
The pressure dependence is usually weak: none at all in the
ideal gas limit (see
Wikipedia:
Speed of sound: Dependence on the properties of the medium).
How long does it take light to travel from the
Sun
to the Earth?
BEHOLD:
where recall "c" is the common physics
for the
vacuum light speed c = 2.99792458*10**8 m/s
(exactly) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.
So about 8 minutes.
If the Sun blew up
right now, we'd live in blissful ignorance for about 8 minutes since
NO signal about the catastrophe
could reach us faster than the
vacuum light speed = 2.99792458*10**5 km/s exactly.
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.
The general formulae
for accumulation of A and exhaustion time t:
For a non-astronomical example---a
real R/P ratio
example---the
proven reserves
of oil (AKA petroleum)
(which is that part of oil resources thought to
be economically extractable:
Wikipedia: Mineral Resources)
is circa 2020
of order 1800 Gbl (i.e., 1800 billion barrels or gigabarrels:
see Wikipedia:
List of Countries by Proven Oil Reserves)
and
the world circa 2020 uses about 30 Gbl/year
(see Wikipedia:
List of Countries by Proven Oil Reserves).
So
R/P ratio is given by
t = R/P ≅ 1800 Gbl/(30 Gbl/year) = 60 years .
If used numbers are treated as hard, then there are
∼ 60 years before all the
oil in the world is gone.
But the numbers are NOT hard.
See also the figure below
(local link /
general link: oil_end.html)
which values from
2003, and so is out-of-date.
Maybe we will NOT
burn the last drop of oil and oil-like fluids,
but move to a
renewable-energy
economy sooner.
What's the distance traveled by light in one year?
BEHOLD:
or more exactly 9.4607304725808*10**15 m.
This, of course, is one light-year (ly).
The exact number is for the Julian year
which has exactly 365.25 days.
Note that a light-year is a unit of distance.
Note also that the vacuum light speed
Light-years
are good natural units
for interstellar distances because:
For their primary base natural unit,
astronomers
use the parsec.
Really, it is just a historical accident that parsecs
are preferred.
But such accidents can never be corrected in
astronomy---the dead hand of the past lies
heavily on us.
We won't go into where parsecs come from right now---we
find out in IAL 2: The Sky.
The parsec is specified thus:
Of course, there are
kiloparsecs (kpc)
(used for INTRAGALACTIC distances since
galaxies are typically a few
kiloparsecs in size)
megaparsecs (Mpc)
(used for INTERGALACTIC distances since nearest-neighbor large
galaxies are typically a few
megaparsecs apart),
and
gigaparsecs (Gpc)
(used for cosmological distances since the
observable universe radius = 14.3 Gpc,
according to the Λ-CDM model).
Finally, the ONLY TWO FORMULAE that the students
using IAL
are probably expected to memorize are those that relate
AMOUNT A, CONSTANT RATE R, and TIME t:
d=vt, and so t=d/v = d/c
= (1.495978707*10**11 m)/(2.99792458*10**8 m/s)
≅ 499.0 s = 8 m, 19.0 s ,
A=Rt, and so t=A/R .
In resource economics,
this exhaustion time ratio is called the
reserves-to-production ratio
or
R/P ratio:
t=R/P=Reserves/Production
which is the time to exhaustion of the reserves if they are known accurately and
production is constant---usually neither is the case.
But the R/P ratio is
still a useful parameter---when it's not
misleading.
Of course, things are NOT as simple as the calculated
R/P ratio value suggests:
d=ct ≅ (3.00*10**8 m/s) * (π*10**7 s)
≅ 9.4*10**15 m ≅ 10**16 m ,
c = 2.99792458*10**5 km/s = 1 ly/year, of course.
Probably,
astronomers should use the
light-year as their
primary base natural unit
for astronomical distances, but, in fact, they only use it as their
secondary base natural unit.
1 parsec = 3.0856776*10**18 cm
= 3.2615638 ly ≅ 3 ly
(see Wikipedia: Parsec:
Equivalencies in other units).
A
A = R * t and t = --- .
R
Special case examples of these are in calculating
distance traveled d at constant speed v in time t AND travel time t at constant speed v over
distance traveled d:
d
d = v * t and t = --- .
v
Form groups of 2 or 3---NOT more---and tackle Homework 1 problems 19--25 on units, natural units, and mathematics.
Discuss each problem and come to a group answer.
Let's work for 5 or so minutes.
The winners get chocolates.
See Solutions 1.
Well, it is by angles and angular coordinates that we locate astronomical objects on the sky and this has been done since ancient times.
But in this IAL, we will just discuss angles and angular measurement.
We will get into locating objects on the sky in IAL 2: The Sky.
In that IAL, we take up the angular coordinate systems the horizontal coordinate system and the equatorial coordinate system, the two most known of the celestial coordinate systems. The equatorial coordinate system is analogous to the geographic coordinate system (AKA longitude and latitude), but pasted on the celestial sphere---the apparent sphere of the sky that surrounds us. In fact, both angular coordinate systems may be about equally old since both are credited in some formulation to Eratosthenes (c.276--c.195 BCE) (see Wikipedia: Armillary sphere: Hellenistic world and ancient Rome; Wikipedia: Geographic Coordinate System: History).
Why specify angles and NOT specify distances too?
Well, we do specify distance when it is useful (and we know it), but it is NOT needed for locating an astronomical object on the sky.
In fact, distances are much harder to determine than angular positions---and this has always been true---techniques for both have vastly improved over the centuries---but the relative difficulty of measuring distances is a constant.
Consider the Ancients (anyone before circa 500 CE by one reckoning) and the Medievals (anyone from circa 500 CE--1500 by one reckoning). 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 methods---which were all that were available to the Ancients and the Medievals.
On the other hand, the Ancients and the Medievals could measure angles fairly accurately---when they weren't being sloppy that is---and today sub-arcsecond accuracy is pretty common: i.e., angles measured to less than 1/3600 of a degree.
Angular measurements are actually quite easily done depending on the accuracy you require, of course.
You can make simple approximate angular measurements with your hand as we will see below in subsection Hand Angle Measurements.
The Babylonians
circa 500--300 BCE
divided the circle
into 360
equal bits (i.e., 360 degrees AKA
360°) as explicated in the
figure below
(local link /
general link: babylonian_360_degrees.html).
There are some finer units that we use occasionally:
Actually, the Ancients did
use arcseconds
for angular diameters
and other very small angles
(see Wikipedia:
Minute and second of arc: Astronomy).
Answer 2 is true too---if you are in the mood.
Actually, there is a sky coordinate system called
the equatorial coordinate system
analogous to longitude and latitude.
We will very briefly discuss it in
IAL 2: The Sky: Location on the Sky and Coordinates.
Just for general astronomical interest, one can make simple angle
measurements with your hands as illustrated by the figure below
(local link /
general link: alien_angular.html).
Recall
that an angular diameter
is the angle
subtended
by the diameter
of a spherical astro-body
(or any spherical body) as measured from some observation point which
is usually the Earth
for astronomy.
I mean as seen from the Earth---which
is what one always means, unless one says otherwise.
Hint: Put your hand at arm's length. Could you cover the
Sun
(which you should NEVER look at with the
naked eye)
or the Moon
with a fist or a finger.
Now the
Sun and the
Moon
have very different sizes.
The fact
that their angular diameters
are nearly equal is
the
great coincidence
which we expand on in the figure below
(local link /
general link: sun_moon_angular.html).
We discuss eclipses
in
IAL 3: The Moon: Orbit, Phases, Eclipses, and More.
If you have angular position, you can have
angular velocity
as illustrated in the figure below
(local link /
general link: angular_velocity.html).
For reference, the complete
Greek alphabet---the
alpha to
the omega---is presented in the
figure below
(local link /
general link: greek_alphabet.html) with
delta in 4th place.
Or from Earth's
perspective, what is the angular
speed of the
Sun
around the
Earth measuring the
Sun relative
to the fixed stars
(i.e., the stars you see in the sky)?
Recall as discussed in the figure above
(local link /
general link: babylonian_360_degrees.html),
the Babylonians
(i.e., the Babylonian astronomers)
may even have chosen the degree size, among other things,
in order to make the angular speed of the
Sun on the sky
about 1 degree per day.
But who knows.
Note the Sun's "orbit" around the
Earth is called a "geometrical orbit", and
is NOT what we mean by orbit, unqualified.
See the definition of orbit
in the figure below
(local link /
general link: orbit_defined.html).
Alas, the French Revolution
(see figure below:
(local link /
general link: tennis_court_oath.html)
that gave us the
Metric System
completely overlooked angular measure, and so we're stuck with
360°
in the circle.
1 degree = 60 arcminutes (')
= 3600 arcseconds ('')
and
1 arcminute (') = 60 arcseconds ('') .
These strange units are because of
the Babylonians
again and their
sexagesimal base system.
We have seconds and minutes in time measurement
because of the
Babylonians too.
Note: An angular diameter
is the angle
subtended
by the diameter
of a spherical astro-body
(or any spherical body) as measured from some observation point which
is usually the Earth
for astronomy.
However, their positional
accuracy/precision was probably
usually much worse than an arcminute
which is the
accuracy/precision
achieved sometimes by
Tycho Brahe (1546--1601),
the greatest pre-telescopic observer
(see Wikipedia: Tycho Brahe:
Observational astronomy).
Question: Why are
angles
on the sky
important in astronomy?
Answer 1 is right. There are lots of other reasons too actually: e.g.,
measurement of angular diameters
and proper motions.
Question: What are the
angular diameters
of the
Sun
and the
Moon?
Answer 3 is right.
The
great coincidence:
causes there to be just marginally
total solar eclipses
and just marginally
annular eclipses.
Note physics
and astronomy
often use Greek letters to
represent standard quantities.
As an example
of an angular velocity calculation,
what is the angular speed of the
Earth
around the
Sun?
The fixed stars are only approximately fixed
over the course of human history.
BEHOLD:
Δθ 360°
-- = ------------- ≅ 1 degree/day .
Δt 365.25 days
Actually it's a little less than 1 degree per day.
Form groups of 2 or 3---NOT more---and tackle Homework 1 problems 26--33 on mathematics and angles.
Discuss each problem and come to a group answer.
Let's work for 5 or so minutes.
The winners get chocolates.
See Solutions 1.
So it's good to have an intro/refresher to plots and functions on plots.
First off, it's good to be able to qualitatively recognize certain function behaviors on (linear) plots.
Some examples are illustrated in the figure below (local link / general link: function_behaviors_plot.html).
Some more examples are illustrated in the
next figure
(local link /
general link: exponential_function_plot.html).
In this course,
we often encounter logarithmic or log plots
which are divided into the categories
log-log plots and
semi-log plots.
On a log axis of a
log plot,
the unit is some power of 10: e.g., 10**(1/2), 10, 10**2,
10**3, etc.
By the by, often and especially in a graphing context,
a factor of 10 is called a dex.
So increasing by a factor 10, 100, 100, etc.
is increasing by, respectively, 1 dex, 2 dex, 3 dex, etc.
The cost of
logarithmic plots is
that functions
are a bit distorted by
linear-scale standards.
But, in fact, there is NOT much cost because
you usually quickly develop an intuitive understanding of them.
Logarithmic plots
are generally useful and
they turn up all the time in
astronomy as we'll see.
Why is explained in the figure below
(local link /
general link: log_log_plot_dj.html).
Ordinary plots
can be called
linear plot
when one has to distinguish them from
log plots
(see Wikipedia: Logarithmic scale).
You do NOT have to know what a
logarithm is
to appreciate log plots.
In fact, you quickly get an intuitive understanding of them.
If you go up one unit, you go up that power of 10.
If both axes are log axes, then the plot is a
log-log plot;
if only one, then the plot is a
semi-log plot.
As special feature of log-log plots is that
they convert
power-law functions
into straight lines.
The explication is in the figure below
(local link /
general link: log_log_plot_wik.html).
Form groups of 2 or 3---NOT more---and tackle Homework 1 problems 30--38 on mathematics and plots.
Discuss each problem and come to a group answer.
Let's work for 5 or so minutes.
The winners get chocolates.
See Solutions 1.
Yours truly wants to make the explication of the topics at the same time completely correct (possible), comprehensible (a reasonable goal), and concise (maybe impossible). But it's hard. So it all gets a bit hairy, but we'll do best to make sense of it.
This section is actually NOT heavily weighted on exams in intro astronomy courses since it is a bit advanced---but it is very important in physics and astronomy.
We consider orbits, gravity, and Newtonian physics further in IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.
THIS SECTION IS UNDER RECONSTRUCTION. THERE IS SIGNIFICANT DUPLICATION TO ELIMINATE. Read as best you can.
In physics jargon, kinematics means the description of motion without consideration of the causes of motion. Kinematics plus the causes of motion is dynamics.
We do, of course, make qualitative use of the concepts of inertial frame, center of mass, force, and acceleration. But we largely leave to section Physics for Orbits (Reading Only) and IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides the discussion of the dynamics orbits.
An orbit is an astro-body's trajectory (relative to some local inertial frame) in a gravitational field or, in other words, under the force of gravity. NO other forces are acting, except for small astrophysical perturbations that are NOT caused by gravity.
For a fuller definition with the special case of greatest interest explicated, see the definition of orbit in the figure below (local link / general link: orbit_defined.html).
An example of astro-body in
orbit is
shown in the figure below
(local link /
general link: iss_orbit.html).
Orbits can be UNBOUND in which case
the
astro-body
is escaping to INFINITY relative to a coordinate origin of interest which is
usually the center of mass
of the system
of
astro-bodies
to which the first named astro-body belongs.
Just to see the size scale of
escape velocities
for Solar System objects
we can take a glance
at Wikipedia: List of Escape Velocities.
Note the escape velocities are usually given in
kilometers per second (km/s) which as we
discussed above in subsection Earth Orbital Velocity is the
natural unit
for the velocity of most macroscopic motions in
space.
Newton's cannonball is a good
way to illustrate what being in
orbit means and the distinction between
BOUND and UNBOUND orbits.
See the figure below
(local link /
general link: newton_cannonball.html)
explicating
Newton's cannonball.
CLOSED orbits are
BOUND orbits where the
trajectory closes on itself.
Due to various
astronomical perturbations,
NO orbit is ever exactly CLOSED, but in many
cases of interest, orbits are CLOSED approximately
to the level of accuracy of interest: e.g., the orbits of
planets
and
moons for many purposes.
Simple
gravitational two-body system
orbits are PLANAR
orbits which
means the orbital trajectory is all in one plane.
See examples of
planes
in the figure below
(local link /
general link: planes_intersecting.html).
Hereafter in this section, we mostly only consider PLANAR CLOSED
orbits,
and so drop the qualifications PLANAR and CLOSED.
Also we'll limit ourselves mostly to orbits in the
Solar System.
This is just to avoid biting off more than we can chew.
But much of discussion generalizes to other
gravitationally-bound
systems: e.g.,
planet-moon systems in general,
planetary systems in general,
binary star systems,
star clusters
(systems of many stars),
galaxies
(giant systems of many stars,
dust, gas, and dark matter)
and
galaxy clusters
(systems of many
galaxies).
INFINITY in this context means so far away from the
system of interest
that NO effects from the
system of interest are significant.
An object must have a sufficiently high velocity to go into
an UNBOUND orbit:
the lower bound on this velocity is called the
escape velocity.
For the
Solar System as a whole,
the useful
celestial frames
(i.e., inertial frame)
is defined by
Solar System center of mass
(which is close to the center of the
Sun, but NOT always inside the
Sun).
To be an inertial frame, it
is, of course, NOT rotating with respect to
observable universe:
i.e., to the bulk mass-energy
of observable universe)
(see Wikipedia:
Inertial frame of reference: General relativity).
Orbits can be BOUND in which case
the astro-body always stays within
some distance of the coordinate origin of interest---unless
a strong
gravitational assists ejects
it to INFINITY.
The planet orbits are NEARLY CIRCLES about the Sun and moon orbits are NEARLY CIRCLES about their respective planets in most cases.
The orbits are all nearly in the same plane---the ecliptic plane as it is called: a point we'll come back to in IAL 2: The Sky.
The ecliptic plane is the plane of the Earth's orbit.
The figure below (local link / general link: moon_clementine.html) illustrates the ecliptic plane.
Asteroids have mainly nearly circular orbits too.
Comets have highly elongated orbits that approximate ellipses. Also comet orbits are NOT confined nearly to the ecliptic plane, but can have any orientation.
Actually, in finer detail, planet,
moons, and
asteroid orbits are all
approximately
elliptical orbits.
The figure below
(local link /
general link: ellipse.html)
explicates ellipses.
More mathematical details about
ellipses
and
elliptical orbits
are given in file
Mathematics file:
ellipse_4.html
and
at site Ellipses and Elliptical Orbits.
Those details are NOT required for this course.
The figure below
(local link /
general link: ellipse_eccentricity.html)
illustrates how ellipse elongation depends on
eccentricity e.
How do the mathematical ellipses connect up with orbits?
Say you have a gravitational two-body system in which one body is MUCH more massive than the other.
In such a gravitational two-body system, the less massive body orbits the more massive body in an ellipse with the more massive body at one focus.
The other focus is just an empty point in space.
We look at Newtonian physics in more detail in section Physics for Orbits (Reading Only) and IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides, but we do NOT derive this ELLIPTICAL ORBIT RESULT result which is actually non-trivial---it gave Isaac Newton (1643--1727) a hard time.
To be general, a
gravitational two-body system
has both bodies
orbiting in ellipses their mutual
center of mass.
The center of mass
is the mass-weighted
average position
of the two bodies.
We discussed center of mass
in file
Mechanics file:
newtonian_physics.html:
Center of Mass.
See the figure below
(local link /
general link: orbit_elliptical_explication.html)
for
gravitational two-body systems
with equal and unequal mass bodies.
The planetary orbits
for the Solar System
are explicated in the figure below
(local link /
general link: sun_planet.html).
For explicating the motions of the
planets, it is
a high accuracy/precision
approximation to assume the
Sun sits unmoving at
Solar System center of mass.
However, it is an approximation as explicated in the figure below
(local link /
general link: solar_system_center_of_mass.html).
You can learn a lot about
orbits in
planetary systems
by looking at the
Orbit videos
below that
(local link /
general link: orbit_videos.html).
The whole observable universe
is self-gravitating
and has a unified evolution for that reason
(among other reasons).
We'll get to the unified evolution in IAL 30: Cosmology.
One needs general relativity
(which we get to in IAL 25: Black Holes)
for the universe as a whole.
Newton's law of universal gravitation
in yours truly's view is
an emergent theory
that emerges from general relativity
in the weak gravity limit
(which includes the
much-smaller-than-the-observable universe limit).
Now every
astro-body
in a planetary system
(e.g., the Solar System)
attracts gravitationally every other
astro-body
of the system because of the long-range nature of
gravity.
If there are only two
gravitationally bound
astro-bodies
in a sufficiently isolated
planetary system (which is a bit unusual),
then their motions are comparatively easy to understand since there is
an exact solution for such
gravitional two-body systems:
i.e., there is a formula you can write down.
As discussed above in subsectionElliptical Orbits,
Center of Mass, and Apsides, the
two bodies orbit their mutual
center of mass
in elliptical orbits.
However, there is NO exact solution in general for
multi-body systems (AKA n-body systems)
with more than 2 bodies (like
most planetary systems:
e.g., the Solar System).
There may be a proof of this is, but yours truly
CANNOT find an explicit statement to that effect.
Special case solutions exist both with and mostly without explicit exact
formulae:
see, e.g.,
Wikipedia:
Three-body problem: Solutions.
See the animations illustrating
three-body systems
in the figure below
(local link /
general link: three_body_system.html).
Re
multi-body systems (AKA n-body systems),
Isaac Newton (1643--1727) was moved to remark:
Perturbation theory
does NOT always work if the system is too difficult.
But often it does.
Newton himself
invented the earliest form
of perturbation theory
(see Nauenberg 1998).
See
Wikipedia: Perturbation theory
and
Wikipedia: Perturbation theory in astronomy.
The
perturbations
of the secondary gravitational sources on virtually all bodies
in orbits
cause the orbits to be NOT exactly
ellipses and
NOT exactly constant in time.
As a result, the Solar System
motions are NOT perfect repeating clockwork although
over the short length of human history
they approximate that.
The
Solar System is, in fact, slowly evolving.
The evolution is actually
chaotic
(see Wikipedia:
Formation and evolution of the Solar System: Long-term stability).
This means that its motions CANNOT be predicted to the far future,
except in a very approximate way.
We will NOT fully describe
chaos, but
the figure below
(local link /
general link: chaos_evolution.html)
gives some explication.
The smaller the body, the less predictable in general because
smaller bodies are more easily affected by the many
weak effects (e.g.,
very weak astronomical perturbations
including light pressure from the Sun).
Some more features (including
astro jargon) for
elliptical orbits are explicated in
the figure below
(local link /
general link: orbit_apsis.html).
The gravitational law is, in fact, different for
the observable universe
as a whole
than for most smaller self-gravitating systems---for
which we use
Newton's law of universal gravitation.
Similarly, on a smaller scale,
planetary systems
are also self-gravitating
and are usually sufficiently isolated that their INTERNAL motions are completely
determined by INTERNAL forces
(which hereafter we assume to be the case)
which is overwhelmingly
the INTERNAL gravitational field.
Thus, they are usually good examples of
celestial frames
(see file
Mechanics file:
frame_hierarchy_astro.html)
since you do NOT need to worry at all about the effect of the
EXTERNAL gravitational field
on the INTERNAL motions since there is NO effect.
The EXTERNAL gravitational field
merely determines the motion of the
center of mass.
In fact, usually if there
are more than two
astro-bodies
in a planetary system,
the motions are immensely complex when examined to
high accuracy/precision.
Unless I am much mistaken, it would exceed the force of human wit
to consider so many causes of the motion at the same time, and to
define the motions by exact laws which would allow of easy calculation.
Actually,
Newton
was mistaken---except for the word "easy"---in regard to the
Solar System
as he went on to show himself.
See Isaac Newton (1643--1727)
in the figure below
(local link /
general link: newton_principia.html).
Without exact solutions (which is overwhelmingly usually the case)
and sometimes with them if they themselves difficult to calculate with,
one is forced to solve
multi-body systems (AKA n-body systems)
by one or other or both of two methods:
Solution by
perturbation theory is explicated
in the figure below
(local link /
general link: orbit_perturbation.html).
In this course, we will often just say
"perturbations do it"
to explain fine details of
self-gravitating systems.
For an examples
of chaos,
astrophysical and non-astrophysical,
see the animation
and videos in
the figure below
(local link /
general link: pendulum_double.html).
Despite being
chaotic, 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.
Form groups of 2 or 3---NOT more---and tackle Homework 1 problems 34--44 on plots and orbits.
Discuss each problem and come to a group answer.
Let's work for 5 or so minutes.
The winners get chocolates.
See Solutions 1.