--- 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).
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
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.
(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) .
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.
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).
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).
MKS units = meters, kilograms, seconds: used in most sciences and engineering CGS units = centimeters, grams, seconds: used in astronomy---very backward of us.
I'll use either as suits my needs.
Here are some useful conversions:
1 kg = 1000 g 1 m = 100 cm . I often use kilometers too: 1 km = 1000 m = 10**5 cm .
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).
Answer 1 is right.
In this class, I ask you to memorize only a very few numbers---maybe only two or three or ...
The first number to memorize is the 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."
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 117 F (see Vegas temperature record).
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).
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 ...
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 center-of-mass free-fall inertial frame (COMFFI 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.
A speed is ratio: distance over time: thus
v = (2πr)/(1 year) = (2 * π * 1.5 * 10^{8} km)/(π * 10^{7} 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).
A range of orbits, including a circular orbit and escape orbits, are illustrated by Newton's cannonball thought experiment (or Gedanken experiment) in the figure below (local link / general link: newton_cannonball.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:
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 ,
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:
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/Productionwhich 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.
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:
d=ct ≅ (3.00*10**8 m/s) * (π*10**7 s) ≅ 9.4*10**15 m ≅ 10**16 m ,
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
c = 2.99792458*10**5 km/s = 1 ly/year, of course.
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:
1 parsec = 3.0856776*10**18 cm = 3.2615638 ly ≅ 3 ly(see Wikipedia: Parsec: Equivalencies in other units).
Of course, there are kiloparsecs (used for INTRAGALACTIC distances since galaxies are typically a few kiloparsecs in size) and megaparsecs (used for INTERGALACTIC distances since nearest-neighbor large galaxies are typically a few megaparsecs apart).
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:
A A = R * t and t = --- . RSpecial 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:
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.
Actually, the Ancients did use arcseconds for angular diameters and other very small angles (see Wikipedia: Minute and second of arc: Astronomy).
Answer 1 is right. There are lots of other reasons too actually: e.g.,
measurement of angular diameters
and proper motions.
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.
Answer 3 is right.
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)?
Δθ 360° -- = ------------- ≅ 1 degree/day . Δt 365.25 daysActually it's a little less than 1 degree per day.
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).
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).
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).
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 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_reference_inertial_frame_basics.html).
Newtonian physics is primarily based on Newton's 3 laws of motion:
(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).
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.
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.
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.
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).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 barycenter (i.e., center of mass) unaffected by the rest of observable universe to high accuracy/precision, except that the barycenter is in free-fall in the external gravitational field due to the rest of observable universe.
So what orbits what? Barycenters orbit barycenters. Often a barycenter is approximately the center of the dominant mass of an isolated gravitationally-bound system: e.g., the Sun is approximately the Solar-System barycenter. For example, the Sun dominates the Solar System. See subsection The Solar-System Barycenter 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.
If the planets suddenly disappeared, the Sun would barely notice.
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.
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).
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.
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).
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.
Let us consider the Sun and a planet as a gravitational two-body system with the Sun much more massive than the planet.
So the Sun is at the center of mass and can be considered unmoving. The situation is illustrated in the figure below (local link / general link: sun_planet.html).
Similarly, the planets all pretty much dominate their moon systems, and so a planet and each of its moon is approximately a gravitational two-body system that are analogous to the Sun-planet systems.
Now, of course, to consider Sun and each planet or planet and each moon as 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.
G * M_1 * M_2 F = --------------- , r**2where the gravitational constant G=6.67430(15)*10**(-11) (MKS units).
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.
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).
The whole universe is self-gravitating and has a unified evolution for that reason among other reasons---which we'll get to 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 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 gravitational perturbations and light pressure from the Sun).
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.