Angular Measurement

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.

A problem:  If d = 20 pc and m = +4, what is M?  (2.5)
And another (more challenging):
If M = 5 and m = 10, how far away is the star?  (100 pc)

Applets

http://lectureonline.cl.msu.edu/~mmp/kap25/Snell/app.htm

http://www.physics.uoguelph.ca/applets/Intro_physics/refraction/LightRefract.html

http://www.physics.metu.edu.tr/~bucurgat/ntnujava/Lens/lens_e.html
For lenses and mirrors, in general - probably more complicated than you need, but this is what is going on in lenses (and mirrors).

Reflection and Refraction

Reflection - light "bouncing" off a reflective surface. This obeys a simple law, the law of reflection!

The incident (incoming) angle equals the reflected angle. Angles are generally measured with respect to a "normal" line (line perpendicular to the surface).

Note that this works for curved mirrors as well, though we must think of a the surface as a series of flat surfaces - in this way, we can see that the light can reflect in a different direction, depending on where it hits the surface of the curved mirror. More to come here.

Refraction revisited:

Refraction is much different. In refraction, light enters a NEW medium. In the new medium, the speed changes. We define the extent to which this new medium changes the speed by a simple ratio, the index of refraction:

n = c/v

In this equation, n is the index of refraction (a number always 1 or greater), c is the speed of light (in a vacuum) and v is the speed of light in the new medium.

The index of refraction for some familiar substances:

vacuum, defined as 1

air, approximately 1

water, 1.33

glass, 1.5

polycarbonate ("high index" lenses), 1.67

diamond, 2.2

The index of refraction is a way of expressing how optically dense a medium is. The actual index of refraction (other than in a vacuum) depends on the incoming wavelength. Different wavelengths have slightly different speeds in (non-vacuum) mediums. For example, red slows down by a certain amount, but violet slows down by a slightly lower amount - meaning that red light goes through a material (glass, for example) a bit faster than violet light. Red light exits first.

In addition, different wavelengths of light are "bent" by slightly different amounts. This is trickier to see. We will explore it soon.

EM Spectrum Chart

Electromagnetic radiation

Light and other types of electromagnetic radiation

You're most familiar with visible light - ROYGBV. This is a tiny sliver of the huge variety of electromagnetic waves given off by things that emit light naturally (stars) and those that generally absorb and reflect or re-emit light (planets, us, etc.)

All forms of these "waves" are called electromagnetic waves, since the wave consists of an electric component and a magnetic component. These waves can be represented on a chart of electromagnetic radiation (waves being emitted) that goes from low frequency waves (with long wavelengths) to high frequency waves (with short wavelength).

All of these waves travel at that same speed in a vacuum (or near vacuum like space) - the speed of light (c).

The product of frequency and wavelength is always the same value - the speed of light.

Speed of light = frequency (f) x wavelength (l)

c = f l

Assignment #2

Assume that g = 10 m/s/s for these problems.  You may need a calculator, though you shouldn't on the quiz.

1.  Consider a ball dropped from rest.  How fast would it be traveling after 2.5 seconds of freefall?

2.  In the problem above, how far would the ball have fallen in this 2.5 seconds?

3.  If you dropped the same ball from the same height on the Moon, would it take a longer or shorter time to hit the surface?

4.  Why are there high altitude directions on boxes of food that has to be boiled?  (Take a look at a box of macaroni and cheese, if possible.)

5.  Thought question.  Knowing what Galileo discovered through his telescope, what do you think was the convincing evidence (for him, or you, if you prefer) that Earth went around the Sun?  In other words, Galileo's telescopic discoveries helped him embrace the Copernican worldview.  Why?

6.  Imagine two stars in space a certain distance apart.  If there distance is tripled (made larger 3 times the original distance), what exactly would happen to the gravitational force between them?

7.  Recall the two short video clips about the inclined planes with the bells.  What was their significance?

8.  Why exactly do two bodies with very different masses fall to the ground with the same acceleration?  (Why would a bowling ball and tennis ball hit the ground at the same time?)

9.  What is "freefall"?  We didn't really discuss this in class, but give it some thought.

10.  Why do astronauts in orbit around the Earth have to exercise so much?


A thought question that does not have an easy answer - really just for you to play with.  It will not be represented on a quiz.  Around the year 1600, virtually no one believed in a heliocentric universe.  By the year 1700, virtually everyone did.  What could have possibly caused such a dramatic change in public opinion?

Tonight's Notes

Galileo Galilei

1609, telescope

·    Moon has craters

·    Way more stars than thought

·    Jupiter has 4 moons – Io, Europa, Ganymede, Callisto…. Jupiter actually has over 60 moons.

·    Saturn has rings!

·    Venus goes through phases (like the Moon)

·    Sun has spots!  (Sunspots)

·    Siderius Nuncius (Starry Messenger)

http://galileo.rice.edu/sci/observations/sunspot_drawings.html

See also Galileo’s book:  Dialogue on Two World Systems (1632)

Newton, 1642-1727

·    Laws of motion – inertia, F=ma, action/reaction

·    Calculus

·    Binomial theorem

·    Alchemy (oops)

·    Rules of optics

·    White light is made of colors (prism)

·    Reflecting telescope (used a mirror)

·    Explained tides

·    Universal gravity

·    Rules of reasoning in philosophy

Principia Mathematica Naturalis Philosophae, 1687

F_g = G m₁ m₂ / d²

Newton’s law of universal gravitation

m₁ = first mass

m₂ = second mass

G = universal constant of gravitation (a very tiny number, 6.67 x 10^-11)

d = distance between masses

This is an INVERSE SQUARE law – meaning that as the distance increases, the force gets smaller at a rate of 1 over the distance squared.  For example, if you double the distance, the new force is ¼ the original.  Triple the distance and the new force is 1/9 the original.

Local gravitation (g)

On Earth, near the surface:

g is approximately 9.8 m/s/s (or around 10 m/s/s).  This means that a freely-falling object increases its speed by roughly 10 m/s with every second of freefall.  Or conversely, if a body is projected upward, it loses roughly 10 m/s upward with each second of travel up.

This looks like this for a falling object:

After 1 second, its speed is 10 m/s.

After 2 seconds, its speed is 20 m/s.

After 3 seconds, 30 m/s

Or if you like equations:

v = g t

The distance that an object falls is a bit trickier to follow.  I will skip the derivation, but it is given by this formula:

d = ½ g t²

Or, d = 5t² , near the surface of the Earth.

So, after 1 second, an object falls 5 m.

After 2 seconds, d = 20 m.

After 3 seconds, d = 45 m.

After 4 seconds, d = 80 m.

Notice how the distance is climbing up exponentially.  If you graphed distance versus time, you’d get a parabola.

Now for Galileo’s odd number’s rule (just for mathematical fun) – see the earlier blog entry.

With increased altitude, g becomes progressively (but slowly) weaker.

On the Moon, local gravity (at the Moon’s surface) is approximately 1/6 that of Earth.

On Venus, it’s around 9/10 that of Earth.

On Jupiter, it’s around 2.5 times that of Earth.

Local gravity (g) can be calculated with this expression:

g = G M_planet / r²

Where G is the same constant as before, M_planet is the mass of your planet and r is the radius of your planet.

So, do all bodies experience the same local gravity?  Why?

FYI: 

http://en.wikipedia.org/wiki/Isaac_Newton

http://en.wikipedia.org/wiki/Galileo

http://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems