Lecture 1: Scientific Notation, Units, Math, Angles, Plots, Motion, Orbits

Don't Panic

Don't you just hate it when newspapers write `a hundred thousand million miles to Arcturus' as if you were just going there to drop the kids off for soccer---Andy Rooney (apocryphal)

Sections

1. Introduction
2. Scientific Notation
3. Units
4. Math
5. Angles and Angular Measurement
6. Plots and Graphs
7. Motion and What Orbits What (Omit)
8. Orbits

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Introduction

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Frequently in introductory astronomy classes some students need a bit of a refresher in math and most students need introduction to some basics that are needed in astronomy.

This lecture is that refresher and introduction.

The math never gets any worse than this lecture---well almost never.

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Scientific Notation

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In astronomy very large and small number turn up all the time. Hence scientific notation is essential. Here is an example:

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

       I got tired writing the html superscript tags and
       they look bad in closely spaced lines.

       So I use an old fortran notation of double asterisks **
         to mean to be raised to.

       Just accept it.

Fortran is not a dead computer language.

Some more examples of scientific notation:

  1) c = 299879245800 cm/s is the

  vacuum speed of light 
   
    which approximately
 
     = 2.998 * 10**10 cm/s  
     = 2.998 * 10**8 m/s   
     = 2.998 * 10**5 km/s . 

  2) amu = 0.000 000 000 000 000 000 000 000 001 6605 kg

              =1.6605 * 10**-27  kg  is the 
           
Atomic Mass Unit (amu)
              which is about the mass of 
              a hydrogen atom which is the lightest atom.

  3)   (9.31 * 10**2)*(2.998 * 10**10)
 
              =9.31 * 2.998 * 10**(2+10)  

              =9.31 * 2.998 * 10**12  ,

              and so exponents add on multiplication

  4) (9.31 * 10**2)/(2.998 * 10**10)

              =(9.31/2.998) * 10**(2-10) 

              =(9.31/2.998) * 10**-8  ,

              and so exponents subtract on division 

Note 3.00 * 10**10 implies that the prefix number is not 3.01 or 2.99.

If the number were more accurately known, it could be 3.004 or 2.996.

In this course, we do NOT worry much about significant figures or quantitative uncertainty estimates.

But they are essential at a higher level.

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Units

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Using CONVENIENT UNITS is the usual rule.

In daily life mph, feet, Fahrenheit degrees are convenient enough---although they are not especially convenient: just reasonably so and, of course, traditional.

But for scientific and engineering purposes one needs units that are adapted to mathematical manipulation.

The main system today is the metric or SI system. There are two main subsets of metric units:

 
     MKS = meters, kilograms, seconds:   
              used in most sciences and engineering
 
     CGS = centimeters, grams, seconds:    
              used in astronomy---very backward of us.
  

I'll use either as suits my needs:

         1 kg = 1000 g 
         1 m = 100 cm   .

         I often use kilometers too:  

         1 km = 1000 m = 10**5 cm  .

There are other funny metric prefixes to pro/demote units by powers of ten. Some of them are rarely used.

``Mega'' symbolized by capital M means million or 10**6, for example.

Alien metric.

Now MKS and CGS are basic reference systems of units that are good for calculation and comparisons of quantities that vary vastly in scale.

For special purposes one often uses units which are particularly suited to the physical system one is dealing with: i.e., one uses CONVENIENT UNITS.

Examples:

1. The standard North American sheet of paper is 11 X 8.5 inches.

   Thus the inch is the natural unit for dealing with the placement of items on a sheet of paper. Centimeters have always seemed pretty useless for dealing with paper sheets---they're too small.

   Alien with convenient units.

2. In dealing with the solar system, the convenient unit of distance is the mean Earth-Sun distance: mean distance because the actual distance varies by about 3.3 %.

   This distance is called the astronomical unit or AU for short:

       1 AU = 1.49597870*10**13 cm = about 1.496*10**13 cm = about 1.5*10**13 cm.
       

   Question:What is the mean Earth-Sun distance in astronomical units?
   1. 1.
   2. 1.496*10**13.
   3. There is too little information to know.

   
   
   Answer 1 is right.

   In this class, I ask you to memorize only a very few numbers---maybe only two or three or...

   The first number to memorize is the Earth-Sun distance in AUs: it's ONE.

   As you can see from the table of planet distances, it is much easier to remember and comprehend solar system astronomical distances in AUs than in kilometers, centimeters, or miles.

3. In dealing with the Earth-Moon system. It is sometimes convenient to know the Earth-Moon distance in Earth radii.

   The Earth's equatorial radius is 6378.140 km: the polar radius is 6356.90 km---the Earth is a bit oblate.

   In equatorial Earth radii, the mean distance of the Moon is

     
       R_Moon= 60.2684 R_Earth_eq = 2.57*10**-3 AU  .
   
       60 Earth radii is the number I remember.
   
       

   One can see that Earth and Moon are both pretty small compared to their separation distance.

   The Earth-Moon system roughly to scale (Cox-16,305).

   [Note: the Moon's distance's varies by about 11 %.

   The eccentricity of the Moon's elliptical orbit is 0.0549. See orbits below.]

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Now there is one quantity whose astrophysically convenient measurement unit might be a bit obscure to you-all: that quantity is temperature.

We will never use the Fahrenheit scale in this class---except to comment on the weather outside.

The Celsius scale is probably familiar to you:

    0 C is the freezing point of water  (32 degrees Fahrenheit).

    100 C is boiling point of water  (212 degrees Fahrenheit).
         (Actually 99.975 C at standard pressure which is 1 atm=1.013*10**5 Pa
         [KWK-208].)

    T_F=T_C*1.8 + 32   is the conversion from Celsius to Fahrenheit.

    

Sometimes we may use Celsius. But usually we'll use the Kelvin temperature scale or, as it is sometimes called, the ABSOLUTE temperature scale.

Kelvin degrees (with symbol K) are the same size as Celsius degrees, but the zero point of the Kelvin scale is absolute zero.

A slight digression.

Temperature is in a sense a measure of random microscopic motion: i.e., atoms or molecules moving about in gases or liquids, or vibrating in solids; also random fields of electromagnetic energy (i.e., light).

[In another sense temperature is a measure of the thermodynamic equilibrium state of matter.]

If that microscopic motion reaches an irremovable minimum (called the zero-point energy in quantum mechanics), then you can't make make any less motion.

You've reached an absolute fundamental lower bound on microscopic motion.

We call that condition absolute zero.

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

And even without reaching it, you can find out easily enough where it is by a various limiting procedures---which are easy enough to do, but we won't discuss them here.

So absolute zero is in fact well known.

It is -273.15 C, in fact, or, as aforesaid 0 K (HRW-429).

The conversion of the Celsius scale to the Kelvin scale and inverse conversion are, respectively:

     T_K=T_C + 273.15

     and

     T_C=T_K - 273.15   .

     

Table of Notable Temperatures

   Notable Temperature     Kelvin (K)    Celsius (C)    Fahrenheit (F)

   absolute zero              0          -273.15          not worth knowing
   coincidence              233.15        -40            -40
   water freezing           273.15          0             32
   human warmish            300            26.85          80.33
   water boiling            373.125        99.975         212
   pure iron melting       1808          1535            who cares
   pure iron boiling       3023          2750              "
   Sun surface             5800           who cares        "
   Sun center             15*10**6          "              "

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Sources: HRW-429, Se-147, and for iron EnvironmentalChemistry.com

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For more details on temperature and the Kelvin scale see an Exposition on the Kelvin scale. Those details though are beyond the scope of this course.

In my view, we should just drop Fahrenheit and Celsius altogether and just use Kelvin all the time. This is my crank idea.

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Math

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Ah, but my Computations, People say,
Reduced the Year to better reckoning?--Nay
'Twas only striking from the Calendar
Unborn To-morrow, and dead Yesterday.

Rubaiyat of of Omar Khayyam, 5th edition, Verse XXXI by Omar Khayyam & Edward Fitzgerald

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 addition, subtraction, multiplication, division, taking a square root, and a little algebra.

Typical kind of MATH we'll encounter is the calculation of speeds or times.

Let us consider some examples.

1. What is the speed of the Earth around the Sun in kilometers/second (km/s)? Why these units? They are the standard ones people use for solar system speeds: km/s. A speed is ratio: distance over time: thus

   Earth's orbital speed.

   Redundantly, we can repeat the calculation.

   
     v=(2 pi r)/(1 year) 
          
      = (2 * pi * 1.5 * 108 km)/(pi * 107 s)
   
      =30 km/s  ,
   
     where
    
     the circumference of a circle is 2*pi*r  ,
   
     r=1.5*108 km  is the AU, of course ,
    
     and
   
     1 year= pi * 107 s to within .5 %  .
   
        

   This last result is just a coincidence: there is nothing deep in it, but it is easy to remember.

   A more exact calculation of the Earth's mean orbital speed gives 29.785 km/s.

   For comparison the escape speed from the Earth is 11.2 km/s (HRW-305).

   The orbital speed of the Earth is determined by the motion of the Earth under gravity, but oddly enough it is almost independent of the Earth's mass (Go3-102).

   But it DOES depend on the Sun's mass as we will discuss in IAWL: Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.

   Question: Of what order is the speed of any body 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 speed of any body 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.

   Head-on asteroid impactor.

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

   BEHOLD

        d=vt,   and so   t=d/v=1.496*10**13 cm / 3*10**10 cm/s
   
                          =500 s=8 min, 20 sec .
   
        

   So about 8 minutes.

   If the Sun blew up right now, we'd live in blissful ignorance for about 8 minutes.

   This problem is one of a general class where you have an amount A and a rate of change R and are asked how long till the amount A is used up.

   BEHOLD

   
        A=Rt,   and so   t=A/R .
   
        

3. For a non-astronomical example, it widely thought that there is about 1000 BILLION BARRELS (i.e., 1000 gigabarrels) of conventional oil reserves left and the world currently uses about 30 BILLION BARRELS of oil per year.

   The end of conventional oil?

   If those numbers are treated as hard, then there are about 1000/30=33 years before all the conventional oil in the world is gone.

   Of course, things are NOT that simple.

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

   2. Much more conventional oil may be found---but this is unlikely since no new giant oil fields have been found since the 1960s.

   3. On the other hand, improved extraction techniques can and have increased recoverable reserves. They could keep on doing that.

   4. Then there is unconventional oil: oil sands and oil shale. These are more expensive to utilize, but they could keep us burning oil for all of the 21st century (e.g., Sm2003-201).

   5. But we might not want to burn all that oil if we get serious about stopping the CO_2 build up in the atmosphere that could be causing climate change.

      Maybe we will not burn the last drop of oil, but move to a hydrogen and renewables economy.

   Still conventional oil might become a lot more expensive as the next decades role by.

   See for example the End of Cheap Oil.

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

          d=ct=(3.00*10**10 cm/s) * (pi*10**7 s)
   
           =9.4*10**17 cm = about 10**18 cm ,
   
          

   or more exactly 9.46053*10**17 cm.

   This, of course, is one light-year (lyr).

   Note astronomers (for reasons I am not prepared to discuss) often use parsecs rather than light-years.

   
          1 parsec=3.0856776*10**18 cm
   
                  =3.26 lyr = about 3 lyr
     
        (Cox-12).
   
          

Finally, probably the only two formulae that the students of these lectures are expected to memorize are those that relate amount, constant rate, and time:

                                          amount
    Amount = Rate * time   and    time = ________
                                           rate

    Special case examples of these are in calculating distance d
    traveled at speed v in time t and travel time t at speed v over
    distance d:

                             d
    d = v * t   and    t =  ___
                             v
 

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Angles and Angular Measurement

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From the Babylonians circa 500 BC (No-39), the circle is divided into 360 equal bits: i.e., 360 degrees.

The Babylonians did NOT say why, but some speculations exist: see Mesopotamia and Angular Measurement.

Alas, the French Revolution that gave us the metric system completely overlooked angular measure, and so we're stuck with 360 degrees.

There are some finer units that we use occasionally:

      1 degree = 60 arcminutes

               = 3600 arcseconds

      and
   
      1 arcminute = 60 arcseconds .

Just for general astronomical interest, one can make simple angle measurements with your hands.

Angle measurements with a hand.

Now everyone's hand is a bit different---we are all unique---but just approximately at arm's length

    1 finger = 1 degree    ,

    1 fist = 10 degrees    ,  and

    1 spread hand = 18 degrees   

       

(Se-18).

These results convenient for judging angles on the sky. Examples of astronomical angles are:

1. the separation between stars,
2. the height of a star above the horizon,
3. the angular diameter of the Moon.

An angular diameter is the angle subtended by the diameter of a spherical body?

Question: What are the angular diameters of the Sun and the Moon?
1. about 10 degrees and 10 degrees, respectively.
2. about 100 degrees and 10 degrees, respectively.
3. about 1/2 degrees and 1/2 degrees, respectively.
4. about 1 arcsecond and 1 arcsecond, respectively.



Answer 3 is right.

The Sun and the Moon have very different sizes: the Sun diameter is about 400 times the Moon's diameter, but the Sun is about 400 times further away than the Moon.

This is the GREAT COINCIDENCE which seemed for long ages to have great cosmic significance, but is now seen just as an accident of solar system formation and evolution.



The upshot is the Sun and the Moon have almost the same angular diameter on the sky: i.e., about 0.5 degrees.

More exactly as seen from the Earth's center using mean distances to Sun and Moon and mean diameters we find:

     di_moon=0.515127 degrees

        and 

     di_sun=0.533121 degrees.

    

Question: Why are angles on the sky important?
1. Angular coordinates are how objects are located on the sky.
2. They are fun.
3. They arn't important.



Answer 1 is right.

There is a sky coordinate system analogous to longitude and latitude that we will very briefly discuss in IAWL Lecture 2: The Sky.

Angles are relatively easy to measure. As we've seen you can measure them approximately with your hand.

Even the Ancients could measure angles fairly accurately---when they weren't being sloppy that is---and today accuracy to better than 1 arcsecond (i.e., subarcsecond accuracy) is pretty common.

But the sky has NO APPARENT DEPTH, except that it's far.

There is no simple way to tell distances by eye or even by simple geometric ways available to the Ancients.

Even today distance measurements are relatively hard---relative to angular measurements.

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If you have angular position, you can have ANGULAR VELOCITY.

For example say that the angular speed is a constant, then the angular speed just equals any change in angle divided by the corresponding change in time:

       Delta theta
       ____________  =   angular velocity  , 

       Delta time


                   where Delta means change in.

       

Angular velocity.
A bit of a digression on Greek letters.

Physics and astronomy often use Greek letters to represent standard items

But html doesn't support Greek letters.

So, except in diagrams, I must spell the letters rather than write them.

For reference, the complete Greek alphabet---the alpha to the omega---is presented below with Delta in 4th place.

The Greek Alphabet: alpha, beta, gamma, ...

Alpha, beta, ... --- do you get it?

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''?

Behold

       Delta theta        360 degrees
       ____________  =  ______________   =  about 1 degree/day  .

       Delta time         365.25 days


       

Actually it's a little less than 1 degree per day.

[The ancient Babylonians may even have chosen the degree size in order to make the angular speed of the Sun about 1 degree per day, but who knows.

For some speculation see Mesopotamia and Angular Measurement.]

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Plots and Graphs

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We often have to show plots and graphs---which are synonyms---in this course.

It's good to qualitatively recognize certain function behaviors on graphs.

Some simple function behaviors.

We also sometimes encounter logarithmic plots or log plots.

You do NOT have to know what a logarithm is to appreciate log plots.

On a log plot the unit is some power of 10: e.g., 10**(1/2), 10, 10**2, 10**3, etc.

An example of a log plot.

Log plots are useful for showing quantities that vary over many orders of magnitude.

On ordinary plots much of the quantity's behavior is off the plot or is squashed down to the horizontal axis.

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Motion and What Orbits What (Omit)

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Just a first discussion now.

We'll look at motion and orbits again when we do a deeper discussion of Newtonian physics and gravity in IAWL: Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.

From a geometrical point of view, any position can be taken to define the origin of a motionless reference frame.

A reference frame just be a set of coordinates covering all space.

Geocentric Alien.

But there are physically preferred frames.

For example, nowadays were almost always say the Earth orbits the Sun even though geometrically the reverse is equally true and we DO sometimes take the GEOCENTRIC perspective.

The Sun's reference frame is physically preferred.

This is because Sun's reference frame is a closer approximation to being an inertial frame than the Earth's frame of reference.

Inertial frames are fundamentally unaccelerated frames.

In modern physics, an acceleration is a change in speed and/or a change in direction.

And it is acceleration which is physically noticeable.

Rest and constant-speed straight-line motion (i.e., unaccelerated motion) are not physical distinct: which is which depends on your inertial frame of reference.

Alien in elevator.

The causes of accelerations are forces.

In fact the definition of force is that it is a physical relationship between bodies that can cause an acceleration.

Acceleration and force.

Because it is accelerated, a rotating frame cannot be exactly an inertial frame.

Newton's laws are defined relative to inertial frame.

It is hard to find an absolutely inertial frame:

1. The Earth frame is inertial for many purposes, but it rotates on its axis and revolves around the Sun. The Earth is accelerated.

2. The frame of the Sun and the fixed stars is more inertial, but they revolve around the center of the Milky Way.

   Actually, orbital period of the Sun about the center of the Milky Way. is about 220 million years or 220 Myr (FK-565).

   The fixed stars are NOT really fixed: ``fixed stars'' is a historical term to distinguish relatively unmoving stars from moving planets.

   Newton posited the fixed stars as being the exactly inertial: we've had to change that assumption since his day.

3. The Milky Way has a complex orbit in the Local Group which is the galaxy cluster it belongs to.

4. The Local Group has its own expansion of the universe (IAWL Lecture 31: Cosmology) are probably truly exactly inertial frames.

   We can, in fact, identify our motion with respect to such a frame using cosmic microwave background (CMB) (see also IAWL Lecture 31: Cosmology) and quasars.

But the Earth frame is often sufficiently inertial and the Sun frame more so.

In the more inertial Sun frame it is the planets that orbit the Sun.

Why is this frame is more inertial?

As we will discuss in IAWL Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides, acceleration is proportional to force and INVERSELY proportional to mass.

The gravitational forces between Sun and planets are equal and opposite by Newton's 3rd law.

But the Sun dominates the mass of the solar system.

The whole planetary system has mass of only 0.134 % of the Sun's mass (Cox-293).

So the Sun is much LESS accelerated than the planets and the Sun frame is much more inertial than their frames.

This has been a long story, but it explains why we say ``the planets orbit the Sun.''

If the Sun suddenly disappeared, the planets would fly away from each other in space and never meet again because the major source of gravity was gone: gravity is proportional to mass.

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

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

The Sun dominates the solar system motions.

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Orbits

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What are the orbital shapes?

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

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

Comets have highly elliptical orbits: i.e., closed curves in a plane that are very elongated. Comet orbits are NOT confined to a plane, but can have any orientation.

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

Ellipse diagram.

There's a mathematical formula for ellipses, of course---but I don't what to shock and awe the students.

Instead let's just consider ellipse eccentricity

Eccentricity is a measure of the stretched-outedness of an ellipse.

Eccentricity diagram.

Again there is a formula connecting eccentricity and ellipse shape, but experience suggests students don't want to know it.

How do these mathematical ellipses connect up with orbits?

Say you have a TWO-BODY GRAVITATING SYSTEM in which one body is MUCH more massive that than the other.

In such a TWO-BODY SYSTEM, the less massive body orbits the more massive body in an ellipse with the the more massive body at one focus.

The other focus is just an empty point in space.

Newtonian physics make this happen.

We'll look at Newtonian physics later in more detail in IAWL Lecture 5: Newtonian Physics, Gravity, Orbits, Energy, Tides, but we won't derive this ELLIPTICAL ORBIT RESULT result which is actually non-trivial---it gave Newton a hard time.

[Really a TWO-BODY GRAVITATING SYSTEM has both bodies orbiting in ellipses their mutual center of mass of the solar system.

The center of mass is the mass-weighted average position of the two bodies. We won't give a formula here.

But if one body is much more massive, it effectively is the center of mass and is at the focus of the other bodies orbit.

This is the case we've just discussed.]

Let us consider the Sun and a planet as a TWO-BODY SYSTEM with the Sun much more massive than the planet.

Sun and planet.

Note that mean distance to the planet r is the average of the distances to perihelion and aphelion.

The eccentricity e turns out to the relative variation of the perihelion and aphelion distances from the mean distance.

For example, if e=0.2, then the perihelion distance is 20 % smaller and the aphelion distance is 20 % larger than the mean distance.

Planets move fastest at perihelion and slowest at aphelion.

The names perihelion and aphelion apply strictly only to orbits about the Sun.

But same special points occur in any elliptical orbit.

For orbits about Earth, one uses perigee and apogee: GEE is derived from the Greek for Earth.

For orbits about a star, one uses periastron and apastron: astron just being derived from Latin for star.

Probably, one could just use perigee and apogee or periastron and apastron generally.

Sometimes people use very specific names for the points: e.g., PERI-JUPITER and AP-JUPITER for orbits about Jupiter.

The Sun so dominates the planetary system that the Sun and each planet is approximately a TWO-BODY SYSTEM.

Similarly, the planets all pretty much dominate their moon systems, and so a planet and each of its moons is approximately a TWO-BODY SYSTEM that are analogous to the Sun-planet systems.

Now, of course, to consider Sun and each planet or planet and each moon as TWO-BODY SYSTEMS is an approximation.

All bodies in the solar system interact by gravity with all others.

Every bit of mass gravitationally attracts every other bit: this attraction is weak for small-mass bodies, but large for large-mass ones. It falls off by an inverse-square law with distance: F proportional to 1/r**2, where r is the distance separating the body centers.

This makes the real interaction immensely complex.

``Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of the motion at the same time, and to define the motions by exact laws which would allow of easy calculation.''

---Isaac Newton circa 1684 in De Motu, quoted from Cohen & Whitman's Newton's Principia (CW-18).

Actually Newton was mistaken in regard to the solar system as he went on to show himself.

Well maybe the calculation isn't easy, but it can be done.

You first solve for the dominant effect on each body using the TWO-BODY SYSTEM approximation.

Then you correct your calculation, by adding on the effects of the other bodies in order of decreasing importance.

Thus, by a series of corrections you approximate the true behavior of the planets and moons more and more closely.

The secondary effects are called perturbations.

Solving a multi-body system using perturbations.

In this course we will often just say ``perturbations do it'' to explain fine details.

The perturbations of the secondary gravitational sources on bodies cause all the orbits to be NOT exactly ellipses and NOT exactly constant in time.

The solar system motions are not a perfect repeating clockwork although over the course of human history they approximate that.

The solar system is in fact slowly evolving.

The evolution is actually chaotic: i.e., this means super-sensitive to initial conditions among other things.

Chaotic evolution.

But for millions or even billions of years the changes in the orbits of the solar system are small and in particular the major bodies of solar system are approximately predictable like clocks if you have a sufficiently sophisticated computer program.

The smaller the body the less predictable in general, because smaller bodies are more easily affected by the many weak effects (e.g., gravity perturbations and light pressure from the Sun).

More mathematical details about ellipses and elliptical orbits are given on an ellipse page.

Those details are NOT required for this course.