Parallax angle and parsec
- distance (in pc) = 1/(parallax angle in arcsec)
Big Dipper --> Polaris --> Cassiopeia
Big Dipper --> Arcturus --> Spica
Summer Triangle - Vega, Deneb, Altair
Newton's law of gravitation - and its nature as an "inverse square law"
Weightlessness
Origin of solar system
(Nothing on Moon or phases - yet......)
Thursday, June 30, 2016
The Moon
The Moon
Highlands - heavy cratered
Mare (maria, pl.) - smooth
Mass of moon = 1/81 Earth mass
gravity of moon - 1/6 Earth gravity
Diameter of moon - 1/4 Earth diameter
Mountain ranges (formed by debris) and valleys
Ridges and high crater rims
Moon rotates on axis at same rate as its rotation
about the Earth
Synodic Period (about 29.5 days for moon)
As a result, same face is always toward Earth
Most locations in sunlight for 15 days (temp = 130 C)
Also in darkness for 15 days (-110 C)
July 20, 1969 - Aldrin and Armstrong walk on moon
(Apollo 11) while Collins orbits
Six moon missions total (Apollo program)
Lunar Surface:
igneous rocks - formed by lava cooling
few sedimentary rocks (settling)
Thick crust (12% of total volume)
in maria, rocks are mainly basalts
in highlands, anorthosites (rare on earth)
some rocks are breccias (mixtures, welded together)
soil - bits of dust and fragments, small glassy
globules
NO WATER - no life.
Dating of moon done by:
Radioactive dating of material brought back
crater dating (which areas are most heavily cratered)
Craters with rays have formed more recently (they
formed over other areas)
General picture:
Moon formed about 4.6 billion years ago
top 100 km or so was molten for about 200 million
years after
From 4.2 to 3.9 billion years ago:
Heavy bombardment by planetesimals
Moon heated up from radioactive elements inside -
volcanism began
lava flowed onto surface
By 3.1 billion years ago, era of volcanism ends
active lunar history ends (here ends similarity with
Earth)
No tectonics or erosion
Interior
crust: light material, with silica-rich mantle
metallic (iron) core
seismically quiet compared to earth (not totally
sure)
minor magnetic field, frozen into lunar rocks
(possibly left over from old molten core)
slow heat flow from core to surface (1/3 of earth’s)
Origin of moon theories
1. Fission - separated from earth
2. Capture - captured by earth
3. Condensation - formed near and simultaneously with
earth
1 - 3 === > probably not
4. interaction of earth with planetesimals which
formed moon
5. ejection of ring when earth was hit by
planetesimal
The Moon
The Moon
Highlands - heavy cratered
Mare (maria, pl.) - smooth
Mass of moon = 1/81 Earth mass
gravity of moon - 1/6 Earth gravity
Diameter of moon - 1/4 Earth diameter
Mountain ranges (formed by debris) and valleys
Ridges and high crater rims
Moon rotates on axis at same rate as its rotation
about the Earth
Synodic Period (about 29.5 days for moon)
As a result, same face is always toward Earth
Most locations in sunlight for 15 days (temp = 130 C)
Also in darkness for 15 days (-110 C)
July 20, 1969 - Aldrin and Armstrong walk on moon
(Apollo 11) while Collins orbits
Six moon missions total (Apollo program)
Lunar Surface:
igneous rocks - formed by lava cooling
few sedimentary rocks (settling)
Thick crust (12% of total volume)
in maria, rocks are mainly basalts
in highlands, anorthosites (rare on earth)
some rocks are breccias (mixtures, welded together)
soil - bits of dust and fragments, small glassy
globules
NO WATER - no life.
Dating of moon done by:
Radioactive dating of material brought back
crater dating (which areas are most heavily cratered)
Craters with rays have formed more recently (they
formed over other areas)
General picture:
Moon formed about 4.6 billion years ago
top 100 km or so was molten for about 200 million
years after
From 4.2 to 3.9 billion years ago:
Heavy bombardment by planetesimals
Moon heated up from radioactive elements inside -
volcanism began
lava flowed onto surface
By 3.1 billion years ago, era of volcanism ends
active lunar history ends (here ends similarity with
Earth)
No tectonics or erosion
Interior
crust: light material, with silica-rich mantle
metallic (iron) core
seismically quiet compared to earth (not totally
sure)
minor magnetic field, frozen into lunar rocks
(possibly left over from old molten core)
slow heat flow from core to surface (1/3 of earth’s)
Origin of moon theories
1. Fission - separated from earth
2. Capture - captured by earth
3. Condensation - formed near and simultaneously with
earth
1 - 3 === > probably not
4. interaction of earth with planetesimals which
formed moon
5. ejection of ring when earth was hit by
planetesimal
Origin of Solar System
Formation of 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)
- Think "figure skater"
• 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 revoluve 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 0.55
Venus 0.7 0.62 0.95 0.82
Earth 1 1 1 1
Mars 1.5 1.88 0.53 0.11
https://www.classzone.com/books/earth_science/terc/content/visualizations/es0401/es0401page01.cfm?chapter_no=visualization
Tuesday, June 28, 2016
HW 3 -- due either this Thursday or NEXT Thursday
So sorry for the delay - feel free to submit it Monday instead of this Thursday.
1. Explain the meaning of Newton's law of universal gravitation, particularly the inverse square part.
2. What does it mean to be "weightless"?
3. A particular star has a parallax angle of 0.2 arcseconds. How far away is it (in parsecs)?
4. Answer these questions based on a star chart for this month:
a. What star/object is directly overhead?
b. What are the 3 stars of the Summer Triangle, and in what constellations do they reside? Note that one is part of Cygnus (often called the "northern cross") - this helps you tell them apart.
c. Draw the star patterns, starting with the Big Dipper that would help you find these objects: Polaris, Cassiopeia, Arturus, Spica
d. What do you notice about the constellations that reside along the ecliptic?
e. What are the brightest stars in tonight's sky?
If the skies are dark enough, go outside and try to locate everything above.
1. Explain the meaning of Newton's law of universal gravitation, particularly the inverse square part.
2. What does it mean to be "weightless"?
3. A particular star has a parallax angle of 0.2 arcseconds. How far away is it (in parsecs)?
4. Answer these questions based on a star chart for this month:
a. What star/object is directly overhead?
b. What are the 3 stars of the Summer Triangle, and in what constellations do they reside? Note that one is part of Cygnus (often called the "northern cross") - this helps you tell them apart.
c. Draw the star patterns, starting with the Big Dipper that would help you find these objects: Polaris, Cassiopeia, Arturus, Spica
d. What do you notice about the constellations that reside along the ecliptic?
e. What are the brightest stars in tonight's sky?
If the skies are dark enough, go outside and try to locate everything above.
Thursday, June 23, 2016
It's 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.
>
Newton is also remembered for his "laws of motion."
Newton, Philosophiae Naturalis Principia Mathematica (1687) Translated by Andrew Motte (1729)
- often called Principia.
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 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.
>
Newton is also remembered for his "laws of motion."
Newton, Philosophiae Naturalis Principia Mathematica (1687) Translated by Andrew Motte (1729)
- often called Principia.
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, June 21, 2016
Additional test info
DON'T FORGET that you are allowed to use a full sheet of notes, front and back - anything you wish to write on it.
In addition to the quiz and homework topics, you may wish to review these ideas from recent classes:
Copernicus' heliocentric world view
Kepler's laws
Galileo and his telescopic discoveries
Optics - how they work
Longitude and latitude
International Date Line and Prime Meridian
Julian vs. Gregorian Calendar
Text references you may wish to review:
1.1, 1.3
2.1, 2.2
3.1, 3.2, 3.3
4.4, 4.6, 4.7, 4.8, 4.9, 4.10
5.2, 5.3, 5.4, 5.5, 5.6
The earlier concepts:
Epicycles
UT/GMT, daylight vs. standard time
Gregorian vs. Julian calendars
Angular measurement basics
EM spectrum questions
Blackbody radiation
Optics basics - see pictures
Apparent magnitude vs. absolute magnitude
Apparent magnitude scale
light-year, light-second, light-minute
speed of light
meter
second
kilogram
celestial sphere (and related points on it)
asterisms vs. constellations
analemma
declination and RA (right ascension) - how they compare to latitude and longitude
equinoxes and solstices
circumpolar stars
international date line
electromagnetic (EM) spectrum
speed = frequency x wavelength
Know:
the brightest objects in this evening's sky
a couple of asterisms
how to find north
some way to remember the speed of light
how the sequence of radiation on the EM spectrum "works"
UT/GMT, daylight vs. standard time
Gregorian vs. Julian calendars
Angular measurement basics
EM spectrum questions
Blackbody radiation
Optics basics - see pictures
Apparent magnitude vs. absolute magnitude
Apparent magnitude scale
light-year, light-second, light-minute
speed of light
meter
second
kilogram
celestial sphere (and related points on it)
asterisms vs. constellations
analemma
declination and RA (right ascension) - how they compare to latitude and longitude
equinoxes and solstices
circumpolar stars
international date line
electromagnetic (EM) spectrum
speed = frequency x wavelength
Know:
the brightest objects in this evening's sky
a couple of asterisms
how to find north
some way to remember the speed of light
how the sequence of radiation on the EM spectrum "works"
Monday, June 20, 2016
Dobsonian telescope instructions FYI
So, if you have never worked with tools and wood before, these may seem a little prohibitive. The basic telescope need:
- a parabolic mirror
- a secondary mirror (and support called a "spider")
- a tube (usually a concrete forming "sonotube")
- a support for the main mirror
- a focuser and eyepiece
- a box that the tube rests and spins on.
On the other hand, if you use Craigslist, you can usually find a decent used one for around $200. I would recommend that you use a scope that has at least an 8" mirror in it. They're not too heavy, and they can be found used for $200-250. Of course, exercise caution when using Craigslist, as the occasional weirdo can make your life a bit scary. Used telescopes, particularly Dobsonians, rarely have anything wrong with them (since there are so few parts), so buying used can save you time and money.
But if you are feeling adventurous......
https://stellafane.org/tm/dob/
http://www.instructables.com/id/Homemade-125-inch-Dobsonian-Telescope/
http://makezine.com/projects/build-a-backyard-dobsonian-telescope/
http://www.sfsidewalkastronomers.org/?page=building-telescopes
- a parabolic mirror
- a secondary mirror (and support called a "spider")
- a tube (usually a concrete forming "sonotube")
- a support for the main mirror
- a focuser and eyepiece
- a box that the tube rests and spins on.
On the other hand, if you use Craigslist, you can usually find a decent used one for around $200. I would recommend that you use a scope that has at least an 8" mirror in it. They're not too heavy, and they can be found used for $200-250. Of course, exercise caution when using Craigslist, as the occasional weirdo can make your life a bit scary. Used telescopes, particularly Dobsonians, rarely have anything wrong with them (since there are so few parts), so buying used can save you time and money.
But if you are feeling adventurous......
https://stellafane.org/tm/dob/
http://www.instructables.com/id/Homemade-125-inch-Dobsonian-Telescope/
http://makezine.com/projects/build-a-backyard-dobsonian-telescope/
http://www.sfsidewalkastronomers.org/?page=building-telescopes
Friday, June 17, 2016
Quiz 2 topics
Epicycles
UT/GMT, daylight vs. standard time
Gregorian vs. Julian calendars
Angular measurement basics
EM spectrum questions
Blackbody radiation
Optics basics - see pictures
Apparent magnitude vs. absolute magnitude
Apparent magnitude scale
Happy weekend, everyone!
UT/GMT, daylight vs. standard time
Gregorian vs. Julian calendars
Angular measurement basics
EM spectrum questions
Blackbody radiation
Optics basics - see pictures
Apparent magnitude vs. absolute magnitude
Apparent magnitude scale
Happy weekend, everyone!
Thursday, June 16, 2016
The beginning of modern astronomy: Copernicus, Galileo, Kepler
Nicolaus Copernicus, 1473 - 1543
http://astro.unl.edu/naap/ssm/animations/configurationsSimulator.html
Galileo Galilei, 1564 - 1642
Galileo and his telescope:
moon craters
moons of Jupiter
phases of Venus
"rings" of Saturn
stars in the Milky Way
sunspots
Speaking of sunspots:
http://galileo.rice.edu/sci/observations/sunspot_drawings.html
Johannes Kepler, 1571-1630
- his laws (shown below) are based on the observations of Tycho Brahe
- his laws (shown below) are based on the observations of Tycho Brahe
http://astro.unl.edu/naap/pos/animations/kepler.swf
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 (P, for period) to go around the Sun in a very peculiar fashion:
a^3 = P^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:
Example problem: Consider an asteroid with a semi-major axis of orbit of 4 AU. We can quickly calculate that its period (P) of orbit is 8 years (since 4 cubed equals 8 squared).
Likewise for Pluto: a = 40 AU. P works out to be around 250 years.
Note that for the equation to be an equality, the units MUST be AU and Earth years.
Cool, for fun:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm
History of Astronomy - 1
Ancient science highlights:
Epicycles
Precession
From class:
http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf
The most important things to get out of this were:
- Epicycles were a very useful way to (wrongly) explain why retrograde motion happened with planets.
- Precession (the wobbling of the Earth) causes us to have different North Stars (or no North Star) at various points over the course of thousands of years. Thus, star maps are not accurate after several hundred years. However, this was not understood until the time of Newton and others.
2000 years later.....
From class:
http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf
The most important things to get out of this were:
- Epicycles were a very useful way to (wrongly) explain why retrograde motion happened with planets.
- Precession (the wobbling of the Earth) causes us to have different North Stars (or no North Star) at various points over the course of thousands of years. Thus, star maps are not accurate after several hundred years. However, this was not understood until the time of Newton and others.
2000 years later.....
Scientific Revolution
N. Copernicus, d. 1543
De Revolutionibus Orbium Celestium
http://astro.unl.edu/naap/ssm/heliocentric.html
If you're even more curious:
http://astro.unl.edu/naap/ssm/animations/configurationsSimulator.html
http://astro.unl.edu/naap/ssm/heliocentric.html
If you're even more curious:
http://astro.unl.edu/naap/ssm/animations/configurationsSimulator.html
Next time:
Galileo Galilei, 1564-1642
Siderius Nuncius
Dialogue on Two World Systems
(J. Kepler, C. Huygens, R. Descartes, et. al.)
Isaac Newton, 1642-1727
Principia Mathematica, 1687
Optics images and notes from last class
REFLECTION - "bouncing" of light from a reflective surface
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.
So - light reflects from mirrors, according to the law of reflection. However, if the mirrors is curved, light still obeys this rule - it just looks a bit different. You have to visualize the curved mirror as a series of little flat mirrors.
So - light reflects from mirrors, according to the law of reflection. However, if the mirrors is curved, light still obeys this rule - it just looks a bit different. You have to visualize the curved mirror as a series of little flat mirrors.
REFRACTION - light changing mediums, such as going from air into glass (or water, plastic, etc.)
Consider a wave hitting a new medium - one in which is travels more slowly. This would be like light going from air into water. The light has a certain frequency (which is unchangeable, since its set by whatever atomic process causes it to be emitted). The wavelength has a certain amount set by the equation, c = f l, where l is the wavelength (Greek symbol, lambda).
When the wave enters the new medium it is slowed - the speed becomes lower, but the frequency is fixed. Therefore, the wavelength becomes smaller (in a more dense medium).
Note also that the wave becomes "bent." Look at the image above: in order for the wave front to stay together, part of the wave front is slowed before the remaining part of it hits the surface. This necessarily results in a bend.
MORE DETAIL, for the mathematically inclined:
The general rule - if a wave is going from a lower density medium to one of higher density, the wave is refracted TOWARD the normal (perpendicular to surface) line. See picture above.
Refraction is much different than reflection. 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, but it causes rainbows and prismatic effects.
Some animation, etc.:
http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html
http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm
http://www.freezeray.com/flashFiles/Refraction2.htm
And all of this helps explain how lenses form images.
As shown and discussed in class, light refracts TOWARD a normal line (dotted line on the left image, perpendicular to surface of lens) when entering a more dense medium.
Note in this convex lens that this direction of bend changes from down (with the top ray) to up with the bottom ray. This is due to the geometry of the lens. Look at the picture to make sure that this makes sense. As a result, the rays will intersect after leaving the lens. An image can form!
The FOCAL LENGTH (f) of a lens (or curved mirror) where the light rays would intersect, but ONLY IF THEY WERE INITIALLY PARALLEL to each other. Otherwise, they intersect at some other point, or maybe not at all (if the object is too close to be focused on)!
Note that your (human) eye lenses are convex - slightly thicker in the middle. Thus, your eyes form "real" images on the retina - upside-down! Unless, of course, the object is too close.
If an image is projected onto a screen, the image is REAL. Convex lenses (fatter in the middle) CAN create real images - the only cases where there are no images for convex lenses are when the object distance (between object and lens) is equal to the f, or when do < f. In the first case, there is NO image at all. In the second case, there is a magnified upright virtual image "inside" the lens.
Concave lenses (thinner in the middle) NEVER create real images and ONLY/ALWAYS create virtual images.
Top image depicts parallel light rays hitting a convex lens and meeting at the "focal point." A real image forms at the focal length of a convex lens, WHEN THE RAYS ARE INITIALLY PARALLEL. People who are farsighted wear convex lenses.
The bottom image depicts parallel light rays hitting a concave lens and diverging. In this case, under all circumstances (regardless of where the object is), only virtual images are formed. These can not be projected onto a screen - rather, they appear to reside "inside" the lens. People who are nearsighted wear concave lenses.
Consider a wave hitting a new medium - one in which is travels more slowly. This would be like light going from air into water. The light has a certain frequency (which is unchangeable, since its set by whatever atomic process causes it to be emitted). The wavelength has a certain amount set by the equation, c = f l, where l is the wavelength (Greek symbol, lambda).
When the wave enters the new medium it is slowed - the speed becomes lower, but the frequency is fixed. Therefore, the wavelength becomes smaller (in a more dense medium).
Note also that the wave becomes "bent." Look at the image above: in order for the wave front to stay together, part of the wave front is slowed before the remaining part of it hits the surface. This necessarily results in a bend.
MORE DETAIL, for the mathematically inclined:
The general rule - if a wave is going from a lower density medium to one of higher density, the wave is refracted TOWARD the normal (perpendicular to surface) line. See picture above.
Refraction is much different than reflection. 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, but it causes rainbows and prismatic effects.
Some animation, etc.:
http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html
http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm
http://www.freezeray.com/flashFiles/Refraction2.htm
And all of this helps explain how lenses form images.
Some animation, etc.:
http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html
http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm
http://www.freezeray.com/flashFiles/Refraction2.htm
And all of this helps explain how lenses form images.
As shown and discussed in class, light refracts TOWARD a normal line (dotted line on the left image, perpendicular to surface of lens) when entering a more dense medium.
Note in this convex lens that this direction of bend changes from down (with the top ray) to up with the bottom ray. This is due to the geometry of the lens. Look at the picture to make sure that this makes sense. As a result, the rays will intersect after leaving the lens. An image can form!
The FOCAL LENGTH (f) of a lens (or curved mirror) where the light rays would intersect, but ONLY IF THEY WERE INITIALLY PARALLEL to each other. Otherwise, they intersect at some other point, or maybe not at all (if the object is too close to be focused on)!
Note that your (human) eye lenses are convex - slightly thicker in the middle. Thus, your eyes form "real" images on the retina - upside-down! Unless, of course, the object is too close.
If an image is projected onto a screen, the image is REAL. Convex lenses (fatter in the middle) CAN create real images - the only cases where there are no images for convex lenses are when the object distance (between object and lens) is equal to the f, or when do < f. In the first case, there is NO image at all. In the second case, there is a magnified upright virtual image "inside" the lens.
Concave lenses (thinner in the middle) NEVER create real images and ONLY/ALWAYS create virtual images.
Top image depicts parallel light rays hitting a convex lens and meeting at the "focal point." A real image forms at the focal length of a convex lens, WHEN THE RAYS ARE INITIALLY PARALLEL. People who are farsighted wear convex lenses.
The bottom image depicts parallel light rays hitting a concave lens and diverging. In this case, under all circumstances (regardless of where the object is), only virtual images are formed. These can not be projected onto a screen - rather, they appear to reside "inside" the lens. People who are nearsighted wear concave lenses.
Tuesday, June 14, 2016
Homework #2
Apologies for the delay in posting this!
1. Consider two stars: one has an apparent magnitude of 2 and the other has an apparent magnitude of -1. Which star is brighter, and by what factor is it brighter? (That is, how many times brighter is it than the dimmer one?)
2. Consider the radio station DC101, which generates a signal operating at 101.1 MHz. (MHz means million Hz, or million vibrations per second.)
a. Where on the EM spectrum would you find this station?
b. What is the frequency, speed, and wavelength associated with this signal? You may have to calculate one of these answers - if you're not sure how to complete the calculation, at least show HOW to set it up.
3. Consider the blackbody radiation curve shown in class (and on the blog). What exactly does this represent and how is it related to star light?
4. These questions on time and space will require a bit of research to complete.
a. What is the current Greenwich Mean Time (GMT) and how is it different from the current time where we are?
b. What is the longitude and latitude of your current location. An approximate value for each is fine.
c. What is the current Julian Day (JD)?
d. What calendar are we currently using in the US?
1. Consider two stars: one has an apparent magnitude of 2 and the other has an apparent magnitude of -1. Which star is brighter, and by what factor is it brighter? (That is, how many times brighter is it than the dimmer one?)
2. Consider the radio station DC101, which generates a signal operating at 101.1 MHz. (MHz means million Hz, or million vibrations per second.)
a. Where on the EM spectrum would you find this station?
b. What is the frequency, speed, and wavelength associated with this signal? You may have to calculate one of these answers - if you're not sure how to complete the calculation, at least show HOW to set it up.
3. Consider the blackbody radiation curve shown in class (and on the blog). What exactly does this represent and how is it related to star light?
4. These questions on time and space will require a bit of research to complete.
a. What is the current Greenwich Mean Time (GMT) and how is it different from the current time where we are?
b. What is the longitude and latitude of your current location. An approximate value for each is fine.
c. What is the current Julian Day (JD)?
d. What calendar are we currently using in the US?
Monday, June 13, 2016
Light part 2
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?
SOME MATHEMATICS FOR YOUR INFORMATION, but not to be tested:
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 pc 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.
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