--- The key point to this whole lecture.
Sections
Think of yours truly as Euclid (fl. 300 BCE) in the figure below (local link / general link: euclid.html).
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.
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).
We will now elucidate that natural unit: the kelvin used by the 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 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 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.
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 can't 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---like these
the chessmen
in the figure below
(local link /
general link: chess_lewis.html).
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 body.
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.
The painting
in the figure below
(local link /
general link: eclipse_annular_antoine_caron.html)
seems to depict an annular eclipse
on a cloudy day.
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 body.
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.
See the insert below (local link / general link: frame_basics.html).
Newtonian physics
is primarily based on
Newton's 3 laws of motion:
The
Newton's 2nd law of motion
is often just referred to
as F=ma.
In fact, the internal forces on a body
do cancel out pairwise and this is why they do NOT affect the motion of the
CM though they certainly
affect the motion of the body parts.
There is whole lot of
Newtonian physics
formalism developed on the basis of the primary bases.
Newtonian physics
is strongly believed to hold exactly in the
classical limit and
to be an emergent theory
from TOE-Plus.
In addition to
Newton's 3 laws of motion
Newtonian physics,
is also primarily based on
Newton's law of universal gravitation (which we discuss below in section Orbits,
subsection Solar System Planetary Orbits),
and other force laws.
(vec F_net_ext) = m(vec a_CM) or (vec a_CM) = (vec F_net_ext)/m
where
"vec" means vector (a quantity with a magnitude and a direction),
vec F_net_ext is net external force,
vec a_CM is CM acceleration,
and
m is mass (i.e., the body's resistance to acceleration).
Actually, there are exceptions to the
Newton's 3rd law
as simply stated.
However, those exceptions are treated by the more general principle of
conservation of momemtum
(see
Wikipedia:
Newton's laws of motion: Newton's third law;
Go-7--8).
Recall the classical limit
is the realm where:
Most of everyday life and most astro-bodes from interstellar medium (ISM) to large-scale structure of the universe are close enough to the classical limit that they obey Newtonian physics to a high accuracy/precision.
More general theories can be applied when Newtonian physics fails: e.g., for atoms and molecules, black holes, and the observable universe.
We give some explication in subsections below of the
Newtonian physics
keywords:
acceleration,
center of mass (CM) ,
force,
mass,
etc.
Formally mass is just defined
as the resistance of
a body to acceleration
relative to an inertial frame
and the body's
gravitational "charge"
(i.e., the strength parameter
of its gravitational effects).
However in the classical limit,
the mass of a body equals the
sum of the rest mass
of baryonic matter particles
(i.e., protons,
neutrons,
and electrons)
that make it up.
Because of this statement,
mass is often defined as the
quantity
of matter
as a shorthand.
Note the rest mass is just
the mass-energy
of existence for
massive particles
(i.e., those particles
with rest mass).
Note also that by the dictate of
quantum mechanics,
subatomic particles
and unperturbed
atoms
and
molecules
of a given type are absolutely identical---they have NO freedom to be different.
So each such particle of a given type has exactly the same
rest mass.
Note also again, there are
massless particles:
the photon being the best known.
But actually,
massless particles have
mass since they have
energy as implied by
mass-energy equivalence E=mc**2.
They do NOT
rest mass since they
do NOT exist
at rest
in inertial frames.
What the heck is center of mass
and why do we need it?
Short answer: To clear the bar.
What about the parts of a body and the internal
forces on a body.
The parts of the body are their own bodies with their own
centers of mass
and their own
net external forces
which include those forces
due to other parts of the whole body.
In the astrophysical realm, there is a huge hierarchy of the bodies that are parts of bodies
all with their own
centers of mass
and all held together by self-gravity
and moving under the force
of external gravity:
pressure-supported
astro-bodies
(e.g., planets,
stars,
etc.),
planet-moon systems,
planetary systems,
multiple star systems,
star clusters,
galaxies,
galaxy groups and clusters,
and
galaxy superclusters.
The pressure-supported
astro-bodies
are held up against collapse under
self-gravity mainly by the
pressure force
and a little by the
centrifugal force
due to rotation---which
is NOT a real force, but
Newton's 1st law of motion
in action.
The other astro-bodies are held up
against collapse under
self-gravity by
rotational kinetic energy---they
keep falling to their
centers of mass, but keep missing---this
really what being in orbit is.
The
video
Sun, earth, moon animation | 0:39 below
in Orbit videos
gives an illustration of a
self-gravitating
planetary system held up
by rotational kinetic energy.
Don't panic,
we'll
NEVER calculate a center of mass---we
just need to grok the
concept
center of mass
and learn how to find it without
calculating it in some simple cases.
The figure below
(local link /
general link: center_of_mass_illustrated.html)
illustrates and explicates
center of mass and how to calculate it---if
you really want to know.
There is NO place else centers of mass
could be given that they are mass-weighted average positions.
So one can find the centers of mass
by inspection
in the figure below
(local link /
general link: center_of_mass_2d.html).
However, there is a simple
empirical method for
finding the
center of mass for
rigid systems.
The method is illustrated in the figure below
(local link /
general link: center_of_mass_hanging.html).
Much of the analysis of motion from
Newtonian physics
requires center of mass
(see subsection Newtonian Physics above).
But to give a specific example, we need
center of mass
in understanding how things are held static from a free pivot point: e.g.,
for hanging objects or balancing them.
To explicate:
To further explicate
Newtonian physics
and inertial frames,
we need to define what we mean by
acceleration
and
force.
An acceleration
is a change in speed AND/OR a change in direction.
These two kinds of change are illustrated in the two figures just below
(local link /
general link: gravity_acceleration_little_g.html;
local link /
general link: newton_2nd_law.html).
To explicate:
Often the center-of-mass
inertial frame
of the bodies in the
local system.
A force can depend on
mass (gravity),
electric charge
(the electromagnetic force),
relative position (gravity,
the electromagnetic force),
velocity
(the magnetic force),
and other things.
Note that Newton's 3 laws of motion
are referenced to inertial frames.
It is just part of their statements just as we gave them above in subsection
Newtonian Physics.
However, inertial frames
are often omitted in
initial presentations of the
Newton's 3 laws of motion
to students.
Actually, all physical laws
are referenced to
inertial frames, except
general relativity
(and maybe thermodynamics in some sense)
as aforesaid in subsection Inertial Frames.
Also as aforesaid in subsection
Inertial Frames,
it is general relativity that tells
us what inertial frames are: i.e.,
free-fall frames.
What "referenced to" means is that the laws do NOT work if NOT applied relative to
inertial frames.
This does NOT mean the physical laws
are wrong somehow since they are explicitly or implicitly formulated as
referenced to inertial frames.
More explication of inertial frames---much more explication---is
given above in subsection Inertial Frames.
What if your
reference frame
is an
non-inertial frame
because it's accelerating relative to a local
inertial frame.
A common case is that your
reference frame
is a rotating frame.
If the acceleration is small enough, then the
non-inertial-frame effects
can just be neglected and you can treat
your reference frame
as an approximate inertial frame.
But if your acceleration is NOT small enough,
you could always just switch
from referencing to
a non-inertial frame
to referencing to an inertial frame.
They are just
frames of reference
after all.
As discussed above in subsection Inertial Frames,
the trick is then to treat your
non-inertial frame
as an inertial frame
by introducing
inertial forces (AKA fictitious forces)
which are NOT real
forces,
but just force-like
quantities in the physical
formulae that
give the effects
of being a
non-inertial frame.
We discuss two inertial forces
below in
subsection Inertial Forces on the Earth's Surface
(at somewhat greater length than in
subsection Inertial Frames above).
We usually treat the surface of the Earth
as an inertial frame.
Newtonian physics would NOT
be much use in everyday life if
we could NOT do so as aforesaid above.
All reference frames
NOT accelerated with respect to the
ground also serve
pretty well as
inertial frames.
But you say we are NOT in free fall
on the surface of the Earth, so how
can we treat the
surface of the Earth as an
inertial frame.
Well, spacecraft
Earth is in
free fall.
The center of mass (CM)
of Earth
is in orbit
in the external
gravitational field
of the Sun,
Moon,
to a much lesser degree other
Solar System
astro-bodies,
and perhaps everything else universe
(or our pocket universe
if that is a true theory).
The Earth's gravitational field
is regarded as an internal
gravitational field
of the
CM
free-fall frame
of the Earth.
There are two complications with
treating the ground as
an inertial frame:
We call this tidal force.
It's a stretching force
that is very weak over short distances.
So it doesn't stretch you and me signficantly, but it stretches the
World Ocean
to give us the tides.
See the figure below
(local link /
general link: tide_earth.html).
Therefore the surface of the Earth
CANNOT be exactly an
inertial frame,
but for most, but NOT all, purposes, it's
approximately an
inertial frame:
i.e., it's inertial enough.
Non-inertial frame
effects can be treated, as discussed in subsection
What if Your Reference Frame is an Non-Inertial Frame?
just above, as
inertial forces (AKA fictitious forces)
which is just formalism for treating these
non-inertial frame
effects and NOT real
forces.
We explicate the
inertial forces
on the surface of the Earth
a bit more in the subsection
Inertial Forces on the Earth's Surface
given just below.
There are two main
inertial forces
on the surface of the Earth:
The centrifugal force is
NOT a real force.
It's
Newton's 1st law of motion
in action.
You are trying to go in a straight line
and need to exert a real force to keep
in rotation.
Effectively,
the centrifugal force is
an outward "force" from a center of rotation
in the rotation's own
rotating reference frame.
The centrifugal force
is the thing that tries to throw you off
playground merry-go-rounds:
see the figure below
(local link /
general link: merry_go_round.html).
The centrifugal force
of the Earth
is zero at the poles
where there is no rotation
and strongest at the
equator
where the velocity
of rotation is
0.4651 km/s relative to the
CM
free-fall frame
of the Earth.
The centrifugal force
effect on the Earth's gravity is below
human perception,
but is quite measurable:
e.g., with a gravimeter.
Given the high velocity
at the equator
equator compared
to playground merry-go-rounds,
you may wonder the
centrifugal force
of the Earth
is so small.
The essential answer is
the angular velocity of the
Earth:
360° per day.
You would NOT notice any
centrifugal force
on playground merry-go-rounds
either if it were going that slowly.
To be more physicsy, the
centrifugal force per unit
mass
ranges from 0 at the poles
to ∼ 0.05 N/kg at the
equator
which causes
Earth's effective gravitational field
vary from ∼ 9.83 N/kg at the poles
to ∼ 9.78 N/kg at the
equator
(see Wikipedia: Gravity of Earth: Latitude).
This means you weigh 0.5 % less at the
equator than at the
poles---easily measurable, but
below human perception.
There are also small variations in the
Earth's gravitational field
due to elevation
and varying geology.
All these variations are easily measured too, but are below
human perception.
subsubsection
The other inertial force
on the surface of the Earth
is the Earth's
Coriolis force).
It is an effect due to motion in a
rotating reference frame.
For striking illustration, see the video
Non-inertial Frames of Reference | 0:47
in below
local link /
general link: mechanics_videos.html.
The Coriolis force
on Earth and
other planets is
explained in the figure below
(local link /
general link: coriolis_force.html).
We discussed the determination of
absolute rotation
(i.e., rotation relative to the
observable universe) above
in the figure
frame_inertial_free_fall.html
item
Absolute Rotation Eplicated.
Here we consider the determination of
absolute
translational motion
of the local-to-Earth
inertial frame
to high accuracy/precision.
To do this requires measuring our
translational motion
relative to the
comoving frame
which would be our
local-to-Earth
inertial frame
if we were NOT embedded in the local mass distribution of
of the large-scale structure of the universe
(i.e., the Solar System,
the Milky Way,
the Local Group,
the Virgo Supercluster,
and the Laniakea Supercluster).
We can actually do this to high
accuracy/precision
thanks to the
CMB dipole anisotropy
and the
Doppler effect
(which we explicate in detail in
IAL 7:
Spectra:
The Doppler Effect).
We explicate how and the results in the figure below
(local link /
general link: cmb_dipole_anisotropy.html).
Orbits
(revolving motions in relative to local
inertial frames
and the observable universe)
are everywhere in
the astrophysical realm.
Remember for an isolated
gravitationally-bound system,
all the
astro-bodies
orbit
their mutual
center of mass)
(i.e., barycenter)
unaffected by the rest of
observable universe
to high accuracy/precision,
except that the
center of mass)
is in free-fall
in the external
gravitational field
due to the rest of
observable universe.
So what orbits what?
centers of mass
(i.e., barycenters)
orbit
centers of mass
(i.e., barycenters).
Often a
center of mass
(i.e., barycenter)
is approximately the center of the dominant mass of
an isolated
gravitationally-bound system:
e.g., the
Sun is approximately
the Solar System center of mass (i.e., barycenter).
For example, the Sun dominates the
Solar System.
See subsection
The Solar-System Center of Mass Inertial Frame
below.
Under RECONSTRUCTION.
General relativity
dictates some correction to this situation.
The 2-body system
will slowly lose kinetic energy
due to energy carried away by
gravitational waves
and will inspiral to
coalescence.
This is typically a process taking
gigayears (Gyr),
and so can be neglected in doing ordinary
celestial mechanics.
However, most
gravitationally-bound systems
consist of multiple astro-bodies,
and so all
orbits will be affected by
astronomical perturbations
and will be complex in fine detail at least.
Some orbits
will be complex on the large scale: e.g.,
co-orbital configuration orbits,
horseshoe orbits,
Kozai mechanism orbits,
and
Lissajous orbits
If an astro-body
system is NOT
isolated, then its members probably execute
trajectories that are
NOT characterizable as simple
orbits.
The Sun's mass
is 99.86 % of
the Solar System
mass
(see Wikipedia: Solar System:
Structure and composition).
This overwhelming dominance of the
Sun's mass
means that to good approximation the
Solar-system barycenter
is the Sun's center.
The gravitational force
the Sun is the
main determinant of the
structure of the Solar System.
It pulls the planets
into their
orbits---which means the
planets are in states
of acceleration.
Recall
Newton's 3rd law:
for every force there is an equal and opposite
force---but note these two
forces do NOT have to be on
the same body, and so
just do NOT just cancel out all the time.
Thus the
planets exert equal
gravitational forces on the
Sun to what the
Sun exerts on them.
So the
Sun should also be
accelerated in the frame of fixed stars.
But, as we will discuss
IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides,
acceleration
is proportional to the force
and INVERSELY proportional to
mass.
So the gravitational forces of the
planets
on Sun,
do NOT affect the motion of the Sun very much.
Note to 1st order
only the Sun's
gravitational force
affects a planet.
Thus, to 1st order
the Sun
and each planet form
gravitational two-body system:
i.e., a
system consisting of only 2
gravitationally interacting bodies.
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 instructor can---if he remembers---do a demonstration with a swirling object.
Recall, the fixed stars are
just the relatively nearby stars
(e.g., those that historically define the
constellations)
that are moving in very similar
orbits
to the Solar System's
orbit
around the Milky Way.
The reference frame
defined by the average motion of fixed stars
is technically NOT quite as good an approximation to
an exact inertial frame
as that defined by the
Solar-system barycenter.
This is becasue the fixed stars are all in their
own free-fall frames
which are slightly different than that of the
Solar-System.
barycenter
inertial frame
However, as discussed in figure
frame_inertial_free_fall.html
(shown above in subsection Inertial Frames),
the
absolute rotation
can be measured with respect to the
fixed stars
to good accuracy/precision
which cannot be done by so easily by measurements internal
to the Solar System.
They can be done using a
Foucault pendulum,
of course, as also discussed in figure
frame_inertial_free_fall.html.
In fact, we often reference motion to
the fixed stars as a traditional
way of meaning relatively to an exact local
inertial frame
for Solar System.
Somewhat longer answer: It is the
mass-weighted average position
of a body and we need it since
Newton's 2nd law of motion
(AKA F=ma)
controls the motion of the
center of mass
of a body via the
net external force on the body.
The centers of mass
for objects of sufficiently high symmetry are the obvious centers of symmetry
as illustrated in the figure below
(local link /
general link: center_of_mass_2d.html).
For objects where
center of mass
CANNOT be found
by inspection,
one can do a calculation
from the formula
for center of mass displayed in
the figure shown above
(local link /
general link: center_of_mass_illustrated.html).
The center of mass can be located
deceptively as shown in the figure below
(local link /
general link: center_of_mass_balancing_bird.html).
Why do we need center of mass
in everyday life?
Now a force is a physical relationship
between bodies or between a body and
force field
(e.g., the gravitational field
and electromagnetic field)
that causes an
acceleration of a body relative
to all inertial frames.
If you know the forces
acting on a body from known force laws, then
physical law
will predict the acceleration
relative to the inertial frame you are using.
The physical law
in the classical limit is
Newton's 2nd law of motion (AKA F=ma).
If you are NOT
in the classical limit,
you have to use
relativistic mechanics
and/or quantum mechanics.
Or you could just ignore the whole thing and get bounced around like in
being in an airplane in
clear-air turbulence.
But sometimes it's NOT convenient to switch from
a non-inertial frame
to an inertial frame.
For example, if you are embedded deeply in a
rotating reference frame,
that is your natural
reference frame for
most purposes.
The Earth's
centrifugal force
is due to Earth's rotation
and it causes an effective reduction to
Earth's gravity.
For the Earth,
the Coriolis force is
NOT noticeable on small scales, but it gives rise to
the vortex motion
of cyclones
(see Wikipedia: Cyclone: Structure)
and anticyclones
(see Wikipedia: Anticyclone).
EOF
There is a whole hierarchy of such isolated
gravitationally-bound systems:
Note that from purely geometrical perspective, motion is all relative.
So you could take an point in space as your origin and an rotation as zero rotation.
In fact, for observational purposes
If you have a completely isolated
gravitationally bound
2-body system,
then Newtonian physics
dictates the that the two
astro-bodies
will orbit their mutual
barycenter in
exact ellipses.
This is illustrated in two figures below
(local link /
general link: orbit_elliptical_equal_mass.html;
local link /
general link: orbit_pluto_charon.html)
This inertial frame
of the Solar-system barycenter
is a
free-fall frame orbiting the
center of mass
of the
Milky Way
which defines the
Milky Way
inertial frame.
Since the Solar System
is an isolated
gravitationally-bound system
to very high accuracy/precision,
the internal motions of the
Solar-System
astro-bodies can be analyzed
to high very accuracy/precision
neglecting the rest of the
observable universe.
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, 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).
For the
Solar System as a whole,
the useful inertial frame
is defined by
Solar System barycenter (i.e., 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 UNBOUND in which case the astro-body is escaping to INFINITY relative to a coordinate origin of interest.
Just to see the size scale of escape velocities for Solar System objects we can take a glance at Wikipedia: List of Escape Velocities.
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., moon systems in general, planetary systems in general, binary star systems, star clusters (systems f many stars), galaxies (giant systems of many stars, dust, gas, and dark matter) and galaxy clusters (systems of many galaxies).
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.
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 ellipses.
The figure below
(local link /
general link: ellipse.html)
explicates ellipses.
More mathematical details about
ellipses
and
elliptical orbits
are given on at site Ellipses and Elliptical Orbits.
Those details are NOT required for this course.
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.
Really, 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 discuss center of mass above in the section
Physics and Inertial Frames.
See the figure below
(local link /
general link: orbit_elliptical_equal_mass.html)
for a
gravitational two-body system
with equal-mass bodies.
See the figure below
(local link /
general link: orbit_circular_large_mass_difference.html)
for a
gravitational two-body system
with unequal-mass bodies, but
NOT sufficiently unequal that one body is effectively at
at rest.
The
Sun
so dominates the Solar System---in
mass and
a physics of motion sense (i.e., in
a dynamical sense)---that the
Sun
and each planet individually
constitutes a
gravitational two-body system
to very good approximation.
In fact, the Sun contains 99.86 % of the
Solar System's
known mass
(Wikipedia: Solar System:
Composition),
and so the Sun's center
is to 1st order at the
Solar System center of mass,
except for the
Sun itself.
Thus, we can consider the
Sun
and any planet as a
gravitational two-body system
to very good approximation
with the Sun
sitting unmoving at the
center of mass,
and thus
at the focus point
of the planet's
elliptical orbit.
The situation is illustrated in the figure below
(local link /
general link: sun_planet.html).
Now, of course, to consider Sun
and each planet or planet and each moon as
gravitational two-body system
is an approximation.
All astro-bodies
interact by gravity with all others.
The two-body effect of gravity
is always an attraction.
The force of attraction is proportional to the product of the masses of the bodies:
and so is grows small as r increases.
The 1/r**2 behavior is an
inverse-square law.
More complex mass distributions can be treated by the law by considering them
as being made up of point masses and adding up in a vector sense
the forces between all the pairs of point masses
with pair members being drawn from the two different bodies.
The vector addition is easy in principle though sometimes difficult in practice.
Double the distance and the force decreases by a factor of 4;
triple the distance and the force decreases by a factor of 9; etc.
The fall-off is illustrated in the figure below
(local link /
general link: function_behaviors_plot.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 illustrated
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 principle
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
an exact analytic solution for such
two-body systems:
i.e., there is a formula you can write down.
As discussed above, 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 analytic 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
Wikipedia: Perturbation theory
and
Wikipedia: Perturabion 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 perturbations
including light pressure from the Sun).
Asteroids
have mainly nearly circular orbits too.
The figure below
(local link /
general link: ellipse_eccentricity.html)
illustrates how ellipse elongation depends on
eccentricity e
and gives the relevant formulae too.
How do the mathematical
ellipses
connect up with orbits?
Which focus
has the massive body and which is an empty point is just
determined by initial conditions in the
formation history of the
gravitational two-body system.
Newtonian physics makes this happen.
There is, in fact, an exact analytic solution (i.e., a
formula) for the
gravitational two-body system
(including those with any masses)
that you can just write down.
No other gravitating system has an exact analytic solution.
But if one body is much more massive, it effectively is the
center of mass
and is at the focus
of the other body's orbit.
Some more features (including
astro jargon) for
elliptical orbits are explicated in
the figure below
(local link /
general link: orbit_apsis.html).
In fact, the situation of the
Sun and the
planets,
is mimicked by each
Sun
and its moon system.
A planet and each
moon
is approximately a
gravitational two-body system
with the planet to 1st order at
sitting unmoving at center of mass, and
and thus
at the focus point
of the moon's
elliptical orbit.
However, the degree to which this description holds for
a planet and its
moons varies considerably.
M_1 * M_2 .
It is also inversely proportional to the square of the distance
between the bodies:
1/r**2
Putting the two behaviors together, one has
Newton's law of universal gravitation:
G * M_1 * M_2
F = --------------- ,
r**2
where the
gravitational constant G=6.67430(15)*10**(-11) (MKS units).
This law holds directly only for
point masses
or spherically-symmetric mass distributions.
The fall-off of gravitational attraction with distance---the
inverse-square law behavior---in one sense is rapid.
We recapitulate why
Sun
and each planet
individually can be approximated to 1st order as a
gravitational two-body system
and why the rest of the
observable universe
has NO affect on the internal motions of
planetary systems
in the figure below
(local link /
general link: two_body_system_unexact.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 usually sufficiently isolated that their internal motions are completely
determined by internal forces
(which hereafter we assume to be the case).
The external gravitational field
merely determines the motion of the
center of mass.
Planetary systems
are, in fact, usually good examples of
celestial frames
(which we discussed above subsection
The Basics of Reference Frames
Relevant to Physics)
sufficiently isolated so that only internal forces determine the internal motions.
In fact, usually if there
are more than two
astro-bodies
in a planetary system,
the motions are immensely complex.
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.