IAL 1: Scientific Notation, Units, Math, Angles, Plots, Physics, Orbits

Don't Panic --- The key point to this whole lecture.


  1. Introduction
  2. Scientific Notation
  3. Units
  4. The Kelvin Scale for Temperature
  5. Math
  6. Angles and Angular Measurement
  7. Plots
  8. Physics for Orbits: Reading Only
  9. Orbits

  1. Introduction

  2. Frequently in introductory astronomy classes, some students need a bit of a refresher in math combined with some of the basic lore needed in astronomy.

    Think of yours truly as Euclid (fl. 300 BCE) in the figure below (local link / general link: euclid.html).

    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.

  3. Scientific Notation

  4. In astronomy, very large and small numbers turn up all the time. Some numbers are even astronomic: e.g.,

    So we need a compact and cogent way to represent such numbers.

    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

          a*10**b ,

    where "a" is the coefficient (or, in more elaborate jargon, the significand or mantissa) and "b" is the exponent (or power). In normalized scientific notation,

          1 ≤ a < 10 .

    Note that yours truly usually uses the old fortran notation of double asterisks ** to mean to "raised to the power of". The explanation for this is in the figure below (local link / general link: alien_fortran.html).

    fortran is NOT ... See the figure below (local link / general link: alien_fortran_short.html).

    Here are some
    examples of scientific notation:

    1.   931 = 9.31*10**2, where 10**2 means 102.

    2.   c = 299792458 m/s = 2.99792458*10**8 m/s is the vacuum light speed. Note c is the universal physics symbol for the vacuum light speed.

      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).

    3.   1 AMU = 0.000 000 000 000 000 000 000 000 001 660 539 066 60(50) kg
                   = [1.660 539 066 60(50)]*10**(-27) kg
                   ≅ [1.660 5]*10**(-27) kg
      is the which is the
      atomic mass unit (AMU) = (1/12) C-12 = [1.660 539 066 60(50)]*10**(-27) kg (see NIST: Fundamental Physical Constants --- Complete Listing).

      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.

      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.

      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.

    4. Generic multiplication and division with scientific notation:
        (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.  
    5. To illustrate multiplication with scientific notation:
        (9.31 * 10**2)*(2.998 * 10**10)
          =9.31 * 2.998 * 10**(2+10)  
          =9.31 * 2.998 * 10**12  .  
    6. To illustrate division with scientific notation:
        (9.31 * 10**2)/(2.998 * 10**10)
          =(9.31/2.998) * 10**(2-10) 
          =(9.31/2.998) * 10**(-8)  .  
    7. A fine point: 3.00 * 10**10 implies that the coefficient is NOT 3.01 or 2.99.

      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.

  5. Units

  6. Now for units.

    1. Convenient Units:

      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).

      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.

    2. The Metric System:

      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).

      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.  

      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).

      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.

    3. Natural Units:

      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.

        Actually, Wikipedia---the supreme authority---is formally more restrictive about the use of the term "natural unit".

        But it allows it to be used for convenient units by implication.

      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.

      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.

    4. The Inch:

      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.

    5. The Astronomical Unit (AU):

      1. Introducing the Astronomical Unit (AU):

        To introduce the astronomical unit (AU), see the figure below (local link / general link: astronomical_unit.html).

      2. Using the Astronomical Unit (AU):

        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.

        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).

    6. The Earth Equatorial Radius R_eq_⊕ = 6378.1370 km:

      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 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.

    7. Solar Units for Stars:

      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.

      Solar units are explicated in the figure below (local link / general link: star_natural_units_solar_units.html).

    8. A Natural Unit for Large Areas: The Earth unit (EU): Reading Only and Just for Fun:

      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."

  7. The Kelvin Scale for Temperature

  8. Now there is one quantity, temperature, whose natural unit in astrophysics may NOT be well known to you-all.

    We will now elucidate that natural unit: the kelvin used by the Kelvin scale.

    1. Fahrenheit, Celsius, Kelvin:

      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).

      The Celsius scale is probably familiar to you. Its defining characteristics:

          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
              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.  
      Usually we'll use the Kelvin scale which is more properly called the thermodynamic temperature scale or the ABSOLUTE temperature scale.

      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).

    2. A Small Digression on Absolute Zero:

      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).

      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.

      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.

      Now for a
      macroscopic sample reaching absolute zero seems impossible, but small enough microscopic samples can reach it.

      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.

      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).

    3. Conversions:

      The conversions of the 3 standard temperature scale are given in the figure below (local link / general link: alien_kelvin.html).

      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.

  9. Math

  10. This course is not---NOT---math intensive, but astronomy is.

    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.

    1. Earth Orbital Velocity:

      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.

      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  ,
        the circumference of a circle is 2πr  ,
        r = 1.5*10**8 km  is the astronomical unit, of course,
        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:

      The orbital speed of the Earth is determined by Newtonian physics and initial conditions.

      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).

        Question: Of what order is the speed of any body much smaller than the Sun orbiting the Sun in the Earth's vicinity in a somewhat circular orbit?

        1. Of the same order as the Earth's speed.
        2. Of order 10 times the Earth's speed.
        3. You don't have enough information to say.

        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.

        Question: Of what order is the relative velocity of any astro-bodies in a nearly circular orbit that could impact the Earth.

        1. Of the same order as the Earth's speed.
        2. Anywhere from about zero (compared to 30 km/s) to of order 60 km/s depending on the direction of the impactor.
        3. You don't have enough information to say.

        Answer 2 is right.

      An impactor coming from more or less behind would have a relatively low relative speed; one coming head on would have a high relative speed approaching of order 60 km/s. See the figure below (local link / general link: earth_impactor_velocity.html).

    2. Circular Orbit Velocity and Escape Velocity in General:

      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).

    3. Low Earth Orbital Velocity and Escape Velocity are Really Fast:

      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).

    4. Light Travel Time from the Sun:

      How long does it take light to travel from the Sun to the Earth?


        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:

      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.

    5. The R/P Ratio for Oil: Reading Only:

      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.

      Of course, things are NOT as simple as the calculated
      R/P ratio value suggests:

      1. The rate of use may change. It may go up with increasing demand from rapidly developing countries---most obviously China---but it must fall eventually.

      2. The proven reserves may change with new calculation methods.

      3. Much more oil may be found---but this is unlikely since the discovery of giant oil fields peaked in the 1960s or 1970s (see Wikipedia: Giant oil and gas fields: Recent and future giants).

      4. On the other hand, improved extraction techniques can increase proven reserves. They often do improve.

      5. Then there is currently, economically unrecoverable oil-like fluids: e.g., shale oil (which is NOT the "shale oil" produced by fracking). These are currently too expensive to utilize: "There are no economically viable ways yet known to extract and process shale oil for commercial purposes." (Wikipedia: shale oil: Reserves and production). However, potentially such fluids could keep us burning "oil" for all of the 21st century.

      6. But we might NOT want to burn all oil and oil-like fluids if we want want to stop the rising carbon dioxide (CO2) abundance in the Earth's atmosphere that is causing global warming.

        Maybe we will NOT burn the last drop of oil and oil-like fluids, but move to a renewable-energy economy sooner.

      7. In any case, oil and oil-like fluids might become too expensive compared to renewable energy sometime soon.

    6. Light Travel Distance in a Year:

      What's the distance traveled by light in one year?


        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:

      1. Nearest neighbor stars (NOT counting binary star systems and other multiple star systems, of course) are typically of order one or a few light-years---but there are vast variations.

      2. There is an instant conversion between distance to an object and lookback time (the time since the light signal from object started toward Earth) provided the object has NOT moved much during the lookback time the vacuum light speed is 1 light-year/year. For example, if we see an object 1 million light-years away, we see it as it once 1 million years ago. Note that in many cases, 1 million years is a negligible in cosmic time since many, but NOT all, processes in cosmic time are much longer than a million years.

      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.

      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).

    7. Finally:

      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 = R * t   and    t =  ---   .
      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 = v * t   and    t =  ---   .

  11. Angles and Angular Measurement

  12. Why angles and angular measurement?

    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).

    1. Why Angles and NOT Distances Too?

      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.

      But even today, as indicated above, distance measurements are relatively hard---relative to angular measurements.

      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.

    2. The Babylonians and Angular Units:

      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).

      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.

    3. Arcminutes and Arcseconds:

      There are some finer units that we use occasionally:

            1 degree  = 60  arcminutes (') 
                     = 3600 arcseconds ('') 
            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).

        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).

    4. Hand Angle Measurements:

      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.

    5. Angular Velocity:

      If you have angular position, you can have angular velocity as illustrated in the figure below (local link / general link: angular_velocity.html).

      physics and astronomy often use Greek letters to represent standard quantities.

      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.

      As an
      example of an angular velocity calculation, what is the angular speed of the Earth around the Sun?

      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 days 
      Actually 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).

  13. Plots

  14. We often have to show plots (AKA graphs) in this course.

    So it's good to have an intro/refresher to plots and functions on plots.

    1. Function Behaviors on Linear 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).

    2. Logarithmic Plots:

      In this course, we often encounter logarithmic or log plots which are divided into the categories log-log plots and semi-log plots.

      You do NOT have to know what a logarithm is to appreciate log plots. In fact, you quickly get an intuitive understanding of them.

      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.

        If you go up one unit, you go up that power of 10.

        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.

      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.

      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).

      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).

  15. Physics for Orbits: Reading Only

  16. In order to understand orbits (treated below in section Orbits), we need to understand a little physics including the part about inertial frames.

    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.


    1. The Basics of Inertial Frames:

      See the insert below (local link / general link: frame_reference_inertial_frame_basics.html).

    2. Newtonian Physics:

      Newtonian physics is primarily based on Newton's 3 laws of motion:

      1. Newton's 1st law of motion: The center of mass (CM) of a body will stay in uniform motion (i.e., unaccelerated motion) relative to all local inertial frames (see subsection Inertial Frames below), unless acted on by a net external force.

      2. Newton's 2nd law of motion: A net acceleration of the CM relative to all local inertial frames is caused by (and only caused by) a net external force. As formula, the 2nd law is

          (vec F_net_ext) = m(vec a_CM)   or (vec a_CM) = (vec F_net_ext)/m 
               "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,
               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.

      3. Newton's 3rd law of motion: For every force there is an equal and opposite force. This pair of forces do NOT have to be on the same body. Counterfactually if they did, NO body CM would ever accelerate at all.

        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.

      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.

      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.

    3. Mass:

      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.

    4. Center of Mass:

      What the heck is center of mass and why do we need it?

      Short answer: To clear the bar.

      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.

      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.

    5. Calculating Center of Mass:

      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.

      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).

      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).

      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).

      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).

      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?

      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:

      1. A static balanced object requires the center of mass to directly ABOVE a pivot point. This is an unstable equilibrium since any perturbation causes tipping.
      2. A static hanging object requires the center of mass to directly BELOW a pivot point. This is a stable equilibrium since any perturbation causes an oscillation that damps out: see, e.g., the damped harmonic oscillator
      3. A resting object on a pivot point is a neutral equilibrium. The object will stay a rest for any orientation it is put in.

    6. Acceleration, Force, and Inertial Frames:

      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).

      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.

      To explicate:

      1. Recall all local reference frames NOT accelerating relative to a local inertial frame are also local inertial frames. But there is usually one inertial frame that is most convenient for analysis of a physical system.

        Often the center-of-mass inertial frame of the bodies in the local system.

      2. By physical relationship, one means that the force depends on the nature of the bodies and the states of the bodies.

        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.

      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.

      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.

    7. What if Your Reference Frame is an Non-Inertial Frame?

      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.

      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.

      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).

    8. Hopefully, An Easy Understanding of Inertial Frames on Earth and Beyond:

      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:

      1. The variation in external gravitational field on the CM free-fall frame of the Earth.

        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).

      2. The surface of the Earth is in rotation (and therefore in acceleration) relative to the CM free-fall frame of the Earth.

        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.

    9. Inertial Forces on the Earth's Surface:

      There are two main inertial forces on the surface of the Earth:

      1. The Centrifugal Force:

        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).

        Earth's centrifugal force is due to Earth's rotation and it causes an effective reduction to Earth's gravity.

        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.


      2. The Coriolis Force:

        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).

    10. The Absolute Translation Motion of the Earth:

      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).

    11. Why Do We Need the Physics for Orbits?

      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.

      There are whole hierarchies of such isolated gravitationally-bound systems:
      1. Low-Earth-orbit artificial satellites orbit the Earth's barycenter.
      2. The Earth's barycenter orbits the Earth-Moon system barycenter.
      3. The Earth-Moon system barycenter orbits the Solar-System barycenter.
      4. The Solar-System barycenter orbits the Milky Way barycenter.
      5. The Milky Way barycenter orbits the Local Group of Galaxies barycenter.

      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.

        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)

      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.

    12. The Solar-System Barycenter Inertial Frame:

      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.

      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.

    13. The Gravity of the Sun:

      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.

    14. What if the Sun Vanished?

      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.

        The moons would stay gravitationally bound to the planets, of course.

      If the planets suddenly disappeared, the Sun would barely notice.

    15. The Reference Frame of the Fixed Stars:

      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.

  17. Orbits

  18. In this section, we consider mostly just the kinematics of orbits.

    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.

    1. What Is an Orbit?

      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.

        INFINITY in this context means so far away from the system of interest that NO effects from the system of interest are significant.

      An object must have a sufficiently high velocity to go into an UNBOUND orbit: the lower bound on this velocity is called the escape velocity.

      Just to see the size scale of escape velocities for Solar System objects we can take a glance at Wikipedia: List of Escape Velocities.

      Orbits can be BOUND in which case the astro-body always stays within some distance of the coordinate origin of interest.

      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.

      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).

    2. What Are the Orbital Shapes?

      The planet orbits are NEARLY CIRCLES about the Sun and moon orbits are NEARLY CIRCLES about their respective planets in most cases.

      The orbits are all nearly in the same plane---the ecliptic plane as it is called: a point we'll come back to in IAL 2: The Sky.

      The ecliptic plane is the plane of the Earth's orbit.

      The figure below (local link / general link: moon_clementine.html) illustrates the ecliptic plane.

      Asteroids have mainly nearly circular orbits too.

      Comets have highly elongated orbits that approximate ellipses. Also comet orbits are NOT confined nearly to the ecliptic plane, but can have any orientation.

      Actually, in finer detail, planet, moons, and asteroid orbits are all approximately 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.

      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?

      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.

      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.

      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.

    3. Elliptical Orbits, Barycenters, and Apsides:

      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.

      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.

      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.

      Some more features (including
      astro jargon) for elliptical orbits are explicated in the figure below (local link / general link: orbit_apsis.html).

    4. Solar System Planetary Orbits:

      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).

      Sun so dominates the Solar System---in a physics of motion sense (i.e., in a dynamical sense)---that the Sun and each planet is approximately a gravitational two-body system.

      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:

        M_1 * M_2 .

      It is also inversely proportional to the square of the distance between the bodies:


        and so is grows small as r increases.

        The 1/r**2 behavior is an inverse-square law.

      Putting the two behaviors together, one has Newton's law of universal gravitation:

               G * M_1 * M_2
        F =   ---------------   ,
      where the gravitational constant G=6.67430(15)*10**(-11) (MKS units).
        This law holds directly for point masses or spherically-symmetric mass distributions.

        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.

      The fall-off of gravitational attraction with distance---the inverse-square law behavior---in one sense is rapid.

      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).

      How is it that the
      Sun and each planet individually can be approximated to 1st order as a gravitational two-body system? The explication is given in the figure below (local link / general link: two_body_system_unexact.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).

    5. Multi-Body Self-Gravitating Systems:

      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.

      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, good examples of center-of-mass free-fall inertial (COMFFI) frames (which we discussed above) sufficiently isolated so that only internal forces determine the internal motions.

      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).

      In fact, usually if there are more than two
      astro-bodies in a planetary system, the motions are immensely complex.

      Re multi-body systems (AKA n-body systems), Isaac Newton (1643--1727) was moved to remark:

      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:

      1. By more or less immense numerical calculations on a computer for general cases. This approach has only be available since the 1940s.

      2. By perturbation theory in which one approaches a high accuracy solution by a series of corrections to the solution to an exact solution for a simplified problem.

        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.

      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.

      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.

      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.

      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).