I handed back the Astronomy quiz without telling you what the maximum number of points was. It was 35. If you need a percentage, take 35 minus the points you missed. Then divide by 35.
Good weekend to ya!
Friday, March 27, 2015
Planet Quest
I decided against the idea of Planet presentation in favor of a "Planet Quest" lab. This will be due in 2 classes - April 6.
Lab 4 - Tour
of the Planets
In preparation for the upcoming Solar System test, I would
like you to determine many interesting tidbits of trivia about our solar
neighbors. This may be useful:
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.
Thursday, March 26, 2015
Star Stuff 2 - the Doppler Effect
See this simple, but effective applet:
http://lectureonline.cl.msu.edu/~mmp/applist/doppler/d.htm
In this simulation, v/vs is the ratio of your speed to the speed of sound; e.g., 0.5 is you, or the blue dot, traveling at half the speed of sound. Note how the waves experienced on one side "pile up" (giving an observer a greater detected frequency, or BLUE SHIFT); on the other side, the waves are "stretched apart" (giving an observer a lower detected frequency, or RED SHIFT).
Play with this for a bit, though it's a little less obvious:
http://falstad.com/ripple/
In astronomy, the red shift is very important historically: Edwin Hubble found that light from distant galaxies (as measured in their spectra) was red shifted, meaning that distant galaxies were moving away from us (everywhere we looked). The conclusion was obvious (and startling): The universe is expanding. Last year, local astrophysicist Adam Riess discovered that the rate of expansion was accelerating.
http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/
It's worth noting that the effect also works in reverse. If you (the detector) move toward a sound-emitter, you'll detect a higher frequency. If you move away from a detector move away from a sound-emitter, you'll detect a lower frequency.
Mind you, these Doppler effects only happen WHILE there is relative motion between source and detector (you).
And of course, they also work for light. That's why we care about them. In fact, the terms red shift and blue shift refer mainly to light (or other electromagnetic) phenomena.
In practice, for astronomy:
v = [ (change in wavelength) / (original wavelength) ] c
Star Stuff 1
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)
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
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
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.
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)
Friday, March 13, 2015
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 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.
>
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):
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.
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 25 m/s/s.
>
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.
Tuesday, March 3, 2015
Kepler's Laws
Kepler's laws of planetary motion
http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf
http://astro.unl.edu/naap/pos/animations/kepler.swf
Johannes Kepler, 1571-1630
Note that these laws apply equally well to all orbiting bodies (moons, satellites, comets, etc.)
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
The image below calls period P (rather than T), but the meaning is the same:
Example problem: 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 (since 4 cubed equals 8 squared).
Likewise for Pluto: a = 40 AU. T works out to be around 250 years.
The applets I referenced::
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://astro.unl.edu/naap/ssm/animations/ptolemaic.swf
http://astro.unl.edu/naap/pos/animations/kepler.swf
Johannes Kepler, 1571-1630
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
The image below calls period P (rather than T), but the meaning is the same:
Example problem: 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 (since 4 cubed equals 8 squared).
Likewise for Pluto: a = 40 AU. T works out to be around 250 years.
The applets I referenced::
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
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