Angular Measurement, etc.

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)

Worth watching.

http://www.youtube.com/watch?annotation_id=annotation_753590&feature=iv&src_vid=jN-FfJKgis8&v=hYf6av21x5c

http://www.youtube.com/watch?v=jN-FfJKgis8

(Thanks, Roy!)

Newton HW etc.

I will collect this homework on Thursday.

1.  Review Kepler's laws.  Consider an asteroid with a 10 AU semi-major orbit. How long will it take to orbit the Sun?  

2.  Two bodies in space are separated by a certain distance.  They experience a gravitational force.  If their distance is increased to five times as great, how will the force change exactly?

3.  Consider an object in freefall.  If released from rest, how fast will it be traveling after 3.5 seconds, assuming that it was dropped near the earth surface?  How far can it fall in this time?  It will be helpful to recall the freefall distance formula:  d = 0.5 gt^2

4.  Why is it harder to breathe at high altitudes?  What does this have to do with gravity?  How does boiling point and cooking time (say, for Mac and cheese) change at high altitudes?

5.  Distinguish between weight and mass,

6.  Find your mass in kg.

7.  Now find your weight in newtons.

8.  How is g different on the Moon?  On Mars?

9.  Why exactly do you suppose the heliocentric system met with such resistance?  Forget about the religious aspect for a moment.  Why would it be difficult to convince folks of its correctness?

Kepler and Newton

First, the applets:

http://www.physics.sjsu.edu/tomley/kepler.html

http://www.physics.sjsu.edu/tomley/Kepler12.html
for Kepler's laws, primarily the 2nd law

http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html
for our discussion on geocentrism and how retrograde motion appears within this conceptual framework

Cool:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm

http://physics.unl.edu/~klee/applets/moonphase/moonphase.html

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Now, the notes.

Johannes Kepler, 1571-1630

Kepler's laws of planetary motion - of course, these apply equally well to all orbiting bodies

1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.

2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.

3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the amount of time to go around the Sun in a very peculiar fashion:

a^3 = T^2

That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:

- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles

- the unit of time is the (Earth) year

e.g. Consider an asteroid with a semi-major axis of orbit of 4 AU. We can quickly calculate that its period of orbit is 8 years.

Likewise for Pluto: a = 40 AU. T works out to be around 250 years.

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Newton's take on this 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 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 2.5 m/s/s.

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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.

Lab question

In your Kepler's laws lab conclusion, address the following question:

To what extent does Toro seem to "obey" Kepler's laws?  To answer this, use the numerical data you acquired in comparing:

a.  How elliptical is it?  (x1 + x2 and y1 + y2)
b.  How well do the areas match up?  (A1 + A2)
c.  How well do the ratios compare?  (T^2 / a^3 for Earth vs. Toro)

It won't work out perfectly, no matter what.  Make sure to address WHY there isn't perfect agreement.  There may be multiple reasons for this.  Discuss.

Also - I only need ONE lab per group.

Kepler's Laws HW

First, the lab is due in TWO classes - next Wednesday.  One lab will be submitted per group.

Before next class, please play with this applet:

http://www.physics.sjsu.edu/tomley/kepler.html


And if you'd like a review on retrograde motion and epicycles (which Kepler's laws made obsolete), play with this:

http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf

Or this:

http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html


Fun to play with:

http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm

Galileo stuff

Galileo Sunspot images:

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

Direct link to video:

http://galileo.rice.edu/sci/observations/ssm_slow.mpg

If you have time, watch as much of this as possible.  It's a NOVA special featuring dramatization of G's life.  You can skip around, just watching the dramatization parts.

http://www.youtube.com/watch?v=Lr_bQs4oXgU

Images

In order:

Ptolemaic system

Copernican system (x2)

N. Copernicus

Tycho Brahe

Johannes Kepler

Galileo Galilei

Isaac Newton

History of Astronomy - part 1

First, some history:  epicycles

http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf

Worldviews:

http://www.stumbleupon.com/su/2jRGYC/dd.dynamicdiagrams.com/wp-content/uploads/2011/01/orrery_2006.swf/

http://www.solarsystemscope.com/

Some background details will be discussed in class. Here are some people and dates of note:


Claudius Ptolemy
90 - 168 CE
Almagest, 150 -- the essential book of ancient Greek astronomy.  Employed a geocentric P.O.V., punctuated with epicycles.  THE source on astronomy for well over 1000 years.

Nicolaus Copernicus
1473 - 1543
De Revolutionibus Orbium Celestium
The "Scientific Revolution" is often thought of as beginning with this book.  It's a major treatise that assumes a heliocentric universe:

"... in the midst of all stands the sun. For who could in this most beautiful temple place this lamp in another or better place than that from which it can at the same time illuminate the whole? Which some not unsuitably call the light of the world, others the soul or ruler. Trismegistus calls it the visible God, the Electra of Sophocles, the all-seeing. So indeed does the sun, sitting on the royal throne, steer the revolving family of stars."

Tycho Brahe
1546 - 1601
Adopted a hybrid cosmos:  still Earth-centered, but all planets orbit the Sun (which itself orbits the Earth).

Johannes Kepler
1571 - 1630
Astronomia Nova
Brahe's assistant who was also a strong Copernican.  Used Brahe's massive data to develop what we now call Kepler's Laws (forthcoming in the next blog post).  The first real appearances of celestial mechanics.

Galileo Galilei
1564 - 1642
Siderius Nuncius
Dialogue on Two Chief World Systems
Discourse on Two New Sciences
So much to say about GG.  First modern scientist?  Maybe that's a stretch, but he rightfully gets credit for marrying mathematics and science.  A short list of accomplishments must include his telescopic accomplishments:  sunspots, phases of Venus, moons of Jupiter, craters on the Moon, stars in the Milky Way, rings of Saturn.  He used these ideas to advance the arguments for Copernicanism (which was ultimately his downfall and cause for condemnation).

Isaac Newton
1642 - 1727
Philosophiae Naturalis Principia Mathematica (1687)
The laws of mechanics - a set of "first principles" from which much of physics can be derived
Law of Universal Gravitation
Optics, reflecting telescope
Calculus
Binomial theorem
Tides
Orbits
much more....

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For Galileo:

http://galileo.rice.edu/

http://galileo.rice.edu/bio/index.html

I also recommend "Galileo's Daughter" by Dava Sobel. Actually, anything she writes is pretty great historical reading. See also her "Longitude."

It is also worth reading about Copernicus and the Scientific Revolution.

For those of you interested in ancient science, David Lindberg's "Beginnings of Western Science" is amazing.

In general, John Gribbin's "The Scientists" is a good intro book about the history of science, in general. I recommend this for all interested in the history of intellectual pursuits.

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More historical information regarding Newton:

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

This is really exhaustive - only for the truly interested.

This one is a bit easier to digest:

http://galileoandeinstein.physics.virginia.edu/lectures/newton.html

We'll return to Newton's gravitation (along with Kepler) shortly.

Not astro, but fairly awesome.

http://blogs.smithsonianmag.com/artscience/2013/10/this-alkaline-african-lake-turns-animals-into-stone/?utm_source=facebook.com&utm_medium=socialmedia&utm_campaign=20131003&utm_content=collageofartsandsciencesalkaline

nice!

http://500px.com/photo/10892381?from=set%2F915124