No viewing tonight

Wednesday night instead.  7:30 on the roof.

Problems to play with - a homework assignment

I would like you to work on these questions next week - to be turned in by the end of the week.  You can start to think about them early if you wish.

Radius of Earth = 6.4 x 10^6 m

Radius of Moon = 1.74 x 10^6 m

Radius of Sun = 6.96 x 10^8 m

Distance from Earth to Sun (on average; one "Astronomical Unit" or AU) = 1.5 x 10^11 m

Distance from Earth to Moon = 3.84 x 10^8 m

Mass of Sun (one "solar mass") = 2 x 10^30 kg

Mass of Earth = 6 x 10^24 kg

Speed of light (in a vacuum) = c = 3 x 10^8 m/s

For constant (or average) speed:  v = d/t

Calculate the following:

1.  Distance between Earth and Moon, in Moon diameters. 

2.  Distance from Earth to Sun, in Earth diameters

3.  Amount of time it takes light to reach the Earth from the Moon, in seconds.  

4.  Amount of time it takes light from the Sun to reach Earth, in minutes

5.  Amount of time required for light to travel from the Sun to Pluto (40 AU from the Sun, on average) in hours

6.  Ratio of Earth diameter to Moon diameter

7.  How far light travels in one minute (LM)

8.  Ratio of Sun diameter to Earth diameter

9.  Distance to nearest star (other than Sun) - find the star and the distance (by looking it up)

10.  Approximately how great a distance could light travel during your lifetime?  (This could be termed a "light-lifespan", if you like.)

11.  If you could travel at Earth's escape velocity (11,200 m/s), how long would it take you to reach the Moon?  How long would it take you to reach the nearest star (see 9 above)?

12.  Ratio of the biggest planet (Jupiter) diameter to Earth (diameter)

Formation of the Solar System

~ 4.6 billion years ago huge cloud of gas and dust started collapsing gravitationally

• As it collapsed it spun faster (conservation of angular momentum)

• No (or little) spin in the perpendicular plane

• Local clusters of dust and gas condensed - protosun formed first

• As material cooled, it condensed but never stopped rotating (rotates still since there’s nothing to stop it)

• Cores probably formed first, then attracted neighboring materials to form:  planetesimal, protoplanet

• Probably not a unique system - there is increasing evidence for the existence of many other planetary systems

• Still an evolving theory

• All planets revolve around the sun in the same direction, but 3 have different directions of rotation (relative to the rest and to the direction of solar system motion) - Uranus, Venus, (Pluto)


The Terrestrial Planets:  Mercury, Venus, Earth, and Mars

Relative Characteristics:

Planet         Distance        Period          Radius          Mass

Mercury      0.4               0.24             0.38             0.055
Venus         0.7               0.62             0.95             0.82
Earth          1                  1                 1                 1
Mars           1.5               1.88            0.53             0.11

The Jovian Planets (gas giants)

Jupiter        5.2               11.9            11.2            318
Saturn        9.5               29.5            9.3              95
Uranus       19                84               4.0             14.6
Neptune     30                165             3.9              17.2

Planet "lab"

I decided against the idea of Planet presentation in favor of a "Planet Quest" lab.  This will be due in 2 classes.

Lab 4 - Tour of the Planets

http://nineplanets.org/

Please answer the following questions, based on your reading and web discovery.  Some questions might have several answers, while the answer to others might be "none of them."

Which planet(s):

1.     Rotates backwards?

2.     Revolves backwards?

3.     Rotates nearly on its side?

4.     Have more than 10 moons?

5.     Have only one moon?

6.     Has an orbit with the greatest inclination to the ecliptic?

7.     Is the furthest planet known to the ancients?

8.     Has a largely methane atmosphere?

9.     Has a nondescript, pale greenish color?

10.   Has a blemish known as the great dark spot?

11.   Has a fine iron oxide regolith?

12.   Is most similar to Earth in its surface gravity?

13.   Has the greatest mass?

14.   Has the smallest diameter?

15.   Have been visited by humans?

16.   Has the strongest magnetic field?

17.   Has rings?

18.   Has sulfuric acid clouds?

19.   Has the tallest mountain in the Solar System (and what is it)?

20.   Has a day longer than its year?

21.   Has been landed on most recently by spacecraft?

22.   Experiences global dust storms?

23.   Has a moon that rotates retrograde (and what is it)?

24.   Was once thought to be a failed star?

25.   Is heavily cratered?

26.   Has moons which are likely candidates for life?

27.   Was hit by a large comet in the last several years?

28.   Is most oblate?

Now for the minor bodies.

1.     Which body is an asteroid with its own orbiting asteroid?

2.     Which moon has erupting volcanoes?

3.     Which body is the largest asteroid?

4.     Approximately how many known asteroids are there?

5.     Approximately how many known Kuiper objects are there?  What is the Kuiper belt?

6.     How large is the Oort Cloud?  What is the Oort Cloud?\

7.   What are the Galilean satellites?

8.     Which moon was the first discovered after the Galilean satellites?

9.     What are Sedna and Eris?

10.     What exactly is Pluto?

Etcetera – Write anything else of interest you have uncovered during this exercise.

Star stuff - 1

Big Distances

Consider light as a means of measurement.  The speed of light (in a vacuum) is:

c = 3E8 m/s

(Which is around 186,000 miles per second or 670 million miles per hour, or nearly 7 times around the Earth's equator in one second.)

Since v = d/t, we see that d = v t.

Using light's speed, we can define a light-second:

1 LS = 3 E10 m

Similarly, a light year:

1 LY = 9.4607 E12 km (or nearly 6 trillion miles)


Angular Measurement

Consider the following convention which has been with us since the
rise of Babylonian mathematics:

There are 360 degrees per circle.
Each degree can be further divided into 60 minutes (60'), each called
an arcminute.
Each arcminute can be divided into 60 seconds (60"), each called an arcsecond.
Therefore, there are 3600 arcseconds in one degree.

Some rough approximations:
A fist extended at arm's length subtends an angle of approx. 10º.
A thumb extended at arm's length subtends an angle of approx. 2º.
The Moon (and Sun) subtend an angle of approx. 0.5º.

Human eye resolution (the ability to distinguish between 2 adjacent
objects) is limited to about 1 arcminute – roughly the diameter of a
dime at 60-m.  Actually, given the size of our retina, we're limited
to a resolution of roughly 3'

So, to achieve better resolution, we need more aperture (ie., telescopes).

The Earth's atmosphere limits detail resolution to objects bigger than
0.5", the diameter of a dime at 7-km, or a human hair 2 football
fields away.  This is usually reduced to 1" due to atmospheric
turbulence.

The parsec (pc)

The distance at which 1 AU subtends an angle of one arcsec (1") is
definite as one parsec – that is, it has a parallax of one arcsec.

For example, if a star has a parallax angle (d) of 0.5 arcsec, it is
1/0.5 parsecs (or 2 parsecs) away.

The parsec (pc) is roughly 3.26 light years.

Distance (in pc) = 1 / d

where d is in seconds of arc.

Measuring star distances can be done by measuring their angle of
parallax – typically done over a 6-month period, seeing how the star's
position changes with respect to background stars in 6 months, during
which time the Earth has moved across its ellipse.

Unfortunately, this is limited to nearby stars, some 10,000.  Consider
this:  Proxima Centauri (nearest star) has a parallax angle of 0.75" –
a dime at 5-km.  So, you need to repeat measurements over several
years for accuracy.

This works for stars up to about 300 LY away, less than 1% the
diameter of our galaxy!
[If the MW galaxy were reduced to 130 km (80 mi) in diameter, the
Solar System would be a mere 2 mm (0.08 inches) in width.]

Apparent magnitude (m) scale

This dates back to the time of Hipparchus who classified things as

bright or small.
Ptolemy classified things into numbers:  1-6, with 1 being brightest.
The brightest (1st magnitude) stars were 100 times brighter than the
faintest (6th magnitude).  This convention remains standard to this
day.  Still, this was very qualitative.

In the 19th century, with the advent of photographic means of
recording stars onto plates, a more sophisticated system was adopted.
It held to the original ideas of Ptolemy

A difference of 5 magnitudes (ie., from 1 to 6) is equivalent to a
factor of exactly 100 times.  IN other words, 1st magnitude is 100x
brighter than 6th magnitude.  Or, 6th magnitude is 1/100th as bright
as 1st mag.

This works well, except several bodies are brighter than (the
traditional) 1st mag.

So….. we have 0th magnitude and negative magnitudes for really bright objects.
Examples:
Sirius (brightest star):  -1.5
Sun:  -26.8
Moon:  -12.6
Venus:  -4.4
Canopus (2nd brightest star):  -0.7
Faintest stars visible with eye:  +6
Faintest stars visible from Earth:  +24
Faintest stars visible from Hubble:  +28

The magnitude factor is the 5th root of 100, which equals roughly
2.512 (about 2.5).

Keep in mind that this is APPARENT magnitude, which depends on
distance, actual star luminosity and interstellar matter.
Here's a problem:  What is the brightness difference between two
objects of magnitudes -1 and 6?

Since they are 7 magnitudes apart, the distance is 2.5 to the 7th power, or 600.
For the math buffs:  the formula for apparent magnitude comparison:
m1 – m2 = 2.5 log (I2 / I1)

The m's are magnitudes and the I's are intensities – the ratio of the
intensities gives a comparison factor.  A reference point is m = 100,
corresponding to an intensity of 2.65 x 10^-6 lumens.

Absolute Magnitude, M

Consider how bright the star would be if it were 10 pc away.  This is

how we define absolute magnitude (M).

It depends on the star's luminosity, which is a measure of its brightness:

L = 4 pi R^2 s T^4

R is the radius of the body emitting light, s is the Stefan-Boltzmann
constant (5.67 x 10-8 W/m^2K^4) and T is the effective temperature (in
K) of the body.

So, a star's luminosity depends on its size (radius, R) and absolute temperature (T).

If the star is 10 pm away, its M = m (by definition).
m – M = 5 log (d/10)

We let d = the distance (in pc), log is base 10, m is apparent
magnitude and M is absolute magnitude.

The center of gravity

A very useful concept in astronomy is Center of Gravity (AKA CG or CM, Center of Mass - they are usually the same point).  

Recall the demo with the mass on a stick.  Same mass, held at a further distance from the "fulcrum", is harder to support.  It twists your wrist more - it requires a greater "torque".

So, what is torque?

Torque - a "rotating" force

T = F L

Note that this is sometimes referred to as "moment" or "leverage".

For an object to be "in equilibrium," not only must the forces be balanced, but the torques must also be balanced.

Consider a basic see-saw, initially balanced at the fulcrum:  See image below.

You can have two people of different weight balanced, if their distances are adjusted accordingly:  the heavier person is closer to the fulcrum.  

Mathematically, this requires that the torques be equal on both sides.

Consider two people, 100 lb and 200 lb.  The 100 lb person is 3 feet from the fulcrum.  How far from the fulcrum must the 200 lb person sit, to maintain equilibrium?

Torque on left = Torque on right

100 (3) = 200 (x)

x = 1.5 feet

We call the "balance point" the center of mass (or center of gravity).  

It is the point about which the object best rotates.

It is the average location of mass points on the object.

It does not HAVE to be physically on the object - think of a doughnut.

The principle is believed to originate with Archimedes (287 - 212 BC).  He is believed to have said, "Give me a place to stand on, and I will move the Earth."

What does this have to do with astronomy?  Well, objects orbit around the common center of gravity of the system.  Typically, this is near the center of the star, but with binary star systems, the CG is at some point between them.

Newton

Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:

All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:

F = G m1 m2 / d^2

or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.

Big G = 6.67 x 10^-11 *, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.  This number, the universal gravitation constant, can be thought of as a way of relating mass and distance to force, and arriving at measurable force values.

(Note that the unit for this quantity is Nm^2/kg^2 -- the result of this is that the unit for force works out to be a newton, which is roughly 1/4 of a pound.)

This is an INVERSE SQUARE law, meaning that:

- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.

Weight

Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):

g = G m(planet) / d^2

Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).

Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.

If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.

The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 25 m/s/s.

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Newton is also remembered for his "laws of motion."

Newton, Philosophiae naturalis principia mathematica (1687) Translated by Andrew Motte (1729)

Newton's 3 laws of motion:

1.  Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.


2.  The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.


3.  To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.


In simpler language:

1.  A body will continue doing what it is doing unless there is REASON for it to do otherwise.  It will continue in a straight line at a constant velocity, unless something changes that motion.  This idea is often referred to as INERTIA.

2.  The second law is trickier:

An unbalanced force (F) causes a mass (m) to accelerate (a).  Recalling that acceleration means how rapidly a body changes its speed (in meters per second per second, or m/s/s):

F = m a

There is a new unit here:  the kg m/s/s - this is called a newton (N)

Note that a larger force gives a larger acceleration.  However, with a constant force - the larger the mass is the smaller the acceleration.  Imagine pushing me on a skateboard vs. pushing a small child with the same force - who would accelerate more rapidly?

3.  To every action there is always opposed an equal reaction.

You move forward by pushing backward on the Earth - the Earth, in turn, pushes YOU forward.

A rocket engine pushes hot gases backward - the gases, in turn, push the rocket forward.

If you fire a rifle or pistol, the firearm "kicks" back on you.