Kepler's laws describe the motion of planets orbiting the Sun. Kepler's First Law states that planets follow elliptical orbits with the Sun at one focus. The more distant the foci, the more elongated the ellipse. Kepler's Second Law says that a line connecting a planet to the Sun sweeps out equal areas in equal times. A planet moves fastest when closest to the Sun. Kepler's Third Law relates the square of a planet's orbital period to the cube of its average distance from the Sun.
Use principles of reflection and refraction to describe how lenses and mirrors work.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Use principles of reflection and refraction to describe how lenses and mirrors work.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
This talk will cover the basics of Universal Design for Learning and show how applying principles of UDL to the design of electronic books can transform plain text documents into learning tools. We will show a examples from CAST’s digital learning environments and compare to current trends in eBook interface design.
All'alba della crisi che stiamo ancora vivendo, una riflessione sui segnali che indicavano un crescente bisogno di stringersi in comunità a rete, con lo scopo di cooperare piuttosto che competere. Un cambio di paradigma che sta accelerando, grazie anche alle giovani generazioni dei nativi digitali.
Astronomy. 1511 Laboratory Manual In studying the physics and chemist.pdferodealainz
Astronomy. 1511 Laboratory Manual In studying the physics and chemistry of our solar system,
we learn more about its origins and how it will evolve in time. In light of the discoveries of extra
solar planets and the stars they orbit around, we can use information we learn about them to
understand more about our own planets to reach a deeper understanding. Calculating the Mass of
the Moon Mass is the dominant factor in the initial formation of a solar system, so we: carefully
study the masses of objects to understand how it has an affect during formation and after the
system is stable over a period of time. When you learned about Kepler's Three Laws of Planetary
Motion and Newton's Law of Gravity, you were provided the tools to understand the
relationships between the orbital properties of the planets and the Sun. Because these physical
laws are universal, they also apply to all orbiting objects. As long as the objects are allowed to
move in a natural way, Kepler's Third Law enables us to calculate the mass of the source of
gravity for the orbit. You are going to use a portion of the ephemeris (position and time) data
collected by the Explorer 35 spacecraft when it orbited the Moon to draw its orbit. From this
drawing you will use the properties of the ellipse (refer to your lab on Kepler's Laws and your
textbook for assistance) to determine its semi-major axis. With data from Table 1 and the semi-
major axis measurement, you will use the equation for Kepler's Third Law to calculate the mass
of the Moon. Activity: 1. Plot the data from Table 1 on the graph. Please note that only the
position data is used for the graph. 2. Use a ruler to find the major and minor axis of the ellipse.
This requires you to use your best judgment to determine where they are located. Recall that the
longest distance in the ellipse is the major axis and the shortest distance is the minor axis. Lay
the ruler across the graph and rotate it around until you find them, then draw each line. The
intersection of the two lines should be the geometric center of the ellipse. 3. The center of the
Moon is located on the graph at the coordinates (0,0) in lunar radii. Since the Explorer 35
spacecraft orbited around the Moon, according to Kepler's 1 ut Law, the Moon must be at one
focus of the ellipse. 4. Note that the scale of the grid on the graph is approximately: 1 Lunar
Radii =10 small grid boxes a 23.7mm.
Astronomy 1511 Laboratery Manua! Table 1 - Explorer 35 Ephemeris Data
5. Find and record the following properties of the ellipse in terms of Lunar Radii: (These
properties require either measurement and calculation or just calculation. You will need your
calculator to convert from millimeters to Lunar Radii.) a. The Semi-Major Axis (a): b. The Semi-
Minor Axis (b): c. Measure the distance from the intersection of the lines for the major and
minor axis to the center of the Moon at (0,0). This distance is labeled, c. The Distance from the
Center of the Ellipse to the Focus (c).
Background MaterialAnswer the following questions after revi.docxwilcockiris
Background Material
Answer the following questions after reviewing the “Kepler's Laws and Planetary Motion” and “Newton and Planetary Motion” background pages.
Question 1: Draw a line connecting each law on the left with a description of it on the right.
planets move faster when close to the sun
planets orbit the sun in elliptical paths
planets with large orbits take a long time to complete an orbit
Question 2: When written as P2 = a3 Kepler's 3rd Law (with P in years and a in AU) is applicable to …
a) any object orbiting our sun.
b) any object orbiting any star.
c)
any object orbiting any other object.
Question 3: The ellipse to the right has an eccentricity of about … a) 0.25
b) 0.5
c) 0.75
d) 0.9
Question 4: For a planet in an elliptical orbit to “sweep out equal areas in equal amounts of time” it must …
a) move slowest when near the sun.
b) move fastest when near the sun.
c) move at the same speed at all times.
d) have a perfectly circular orbit.
Question 5: If a planet is twice as far from the sun at aphelion than at perihelion, then the strength of the gravitational force at aphelion will be as it is at perihelion.
a) four times as much
b) twice as much
c) the same
d) one half as much
e) one quarter as much
Kepler’s 1st Law
If you have not already done so, launch the NAAP
Planetary Orbit Simulator
.
·
Tip:
You can change the value of a slider by clicking on the slider bar or by entering a number in the value box.
Open the Kepler’s 1st Law tab if it is not already (it’s open by default).
· Enable all 5 check boxes.
· The white dot is the “simulated planet”. One can click on it and drag it around.
· Change the size of the orbit with the semimajor axis slider. Note how the background grid indicates change in scale while the displayed orbit size remains the same.
· Change the eccentricity and note how it affects the shape of the orbit.
Be aware that the ranges of several parameters are limited by practical issues that occur when creating a simulator rather than any true physical limitations. We have limited the semi-major axis to 50 AU since that covers most of the objects in which we are interested in our solar system and have limited eccentricity to 0.7 since the ellipses would be hard to fit on the screen for larger values. Note that the semi-major axis is aligned horizontally for all elliptical orbits created in this simulator, where they are randomly aligned in our solar system.
· Animate the simulated planet. You may need to increase the animation rate for very large orbits or decrease it for small ones.
· The planetary presets set the simulated planet’s parameters to those like our solar system’s planets. Explore these options.
Question 6: For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit?
Question 7: Create an orbit with a = 20 AU and e = 0. Drag the planet first to the far left of the ellip.
Exercise 1Using the data above in Table 1, make a plot of right .docxrhetttrevannion
Exercise 1
Using the data above in Table 1, make a plot of right ascension versus declination on your printed out Milky Way Globular Clusters Distribution Graph (Diagram 1-the top plot). RA is along the x-axis and goes from 0 to 24 hours, Dec is on the y-axis and goes from +90 to 0 to –90 degrees.) Insert the plot into your lab report with your signature and date.
You will type your answers to the below questions in your lab report and then scan/photo your graph(s) and insert them into your lab document. Again, it would be helpful to review the Exploration from Module 1: “Math Primer for Astronomy” (note this contains link for a free online scientific calculator). There are also good math examples in the Appendix of our eText.
Would you describe the distribution of clusters on the plot as random, or is there a pattern (explain your answer)?
Now look at your plot and point in the direction in which you see most of the globular clusters. This is the general direction of the Galactic Center. Estimate the center of the distribution of the globular clusters. Also estimate (no calculation required — just an educated estimate) the accuracy of determining this center. You have now determined the rough center of our Galaxy!
RA = ____________________ ± ________________
Dec = ____________________ ± ________________
Shapely was correct in thinking that the distribution of globular clusters could reveal something about the Galaxy as a whole. He went one step further. He used the locations of the globular clusters to determine the distance to the Galactic Center. His result was surprisingly accurate and differed from the modern value by less than 10%. So, let’s follow in his footsteps.
The next step is to determine the distance to the clusters. Shapely did this by using RR Lyrae stars. These are variable stars, which have a relatively narrow range of luminosities. From the difference between the apparent magnitudes (measured from his photographic plates) and the absolute magnitudes (calculated from the luminosities), he calculated the distances in parsecs to the star (via: m - M = 5log10(d) + 5). So now we have the distances and the directions of the globular clusters and we can determine the 3-dimensional distributions of the globular clusters relative to us.
However, we will use a different coordinate system that is based on galactic latitude and longitude rather than RA and Dec. The plane of the Galaxy is designated as “0 latitude”. Why would we want to do this? RA and Dec is a messy coordinate system that depends on our orientation in space and the earth’s rotation around its axis. The system based on galactic latitude and longitude is therefore simpler. However, it means that we have to transform the measured RA and DEC positions of the globular clusters and galactic latitude and longitude. To simplify things even further, let’s express the galactic latitude and longitude in terms of x, y, and z coordinates. The advantage of this is that x,.
Module 02 – Kepler’s Laws Lab / Understanding Planetary Motion
Johannes Kepler, a 17th Century astronomer and mathematician, published three laws of planetary motion that improved upon Copernicus’s heliocentric model. These laws were made possible by years of accurate planetary measurement collected by Kepler’s predecessor, Tycho Brahe. Kepler’s laws were a radical change from previous astronomical models for the Solar System which maintained the ancient Greek idea of perfect circular motion. With the Stellarium planetarium software, we are able observe the orbit of the planets and test some of his ideas.
Background Question – Describe Kepler’s three laws of planetary motion.
Object: Explain the purpose of this laboratory assignment in your own words. What do you think you will accomplish or learn from this exercise?
Hypothesis: Write a simple hypothesis connected to Kepler’s laws of motion that you will be able to test using the Stellarium software (for example, if Kepler’s laws are correct, Mercury should move fastest in its orbit when it is closest to the Sun).
Procedure
1) Open the Stellarium software. Open the location window (F6) and change the planet to Solar System Observer. This will change our observing location to a position outside our Solar System.
2) Open the Sky and Viewing options window (F4). Under the “Sky” tab, uncheck the Atmosphere, Stars, and Dynamic eye adaption. Check “Show planet markers” and “Show planet orbits”.
3) Select the Landscape tab and uncheck “Show ground”.
4) Open the Search window (F3) and enter in the Sun. The view should shift such that the Sun is in the center of the screen.
5) With the mouse wheel, zoom in toward the Sun and you should be about to see the orbits and position of each of the planets. If you left click on one of the planets, then only that particular planet’s orbit will be displayed. With the time control at the bottom right, accelerate the flow of time until you see the planets moving in their orbits.
Q1. List the visible planets in order of increasing distance from Sun.
Q2. Are the planets moving at the same speed? If not, which planet is the fastest and what planet is the slowest
6) Zoom down until you see Mercury orbit. Left click on Mercury so you only see Mercury’s orbit and information on the left.
Q3. Is Mercury orbit perfectly circular or is it slightly egg shaped?
Q4. Is the Sun at the exact center of Mercury orbit?
7) Click somewhere off a planet so all the planets’ orbits are displayed. Zoom out until you see the orbit of Mars. Open the Search window and type in 2P/Encke. Stellarium will center on a comet that has a very elliptical orbit. Increase the flow of time enough so you can see Comet Encke move in orbit around the Sun
Q5. When does Encke move the fastest? Is this in agreement with Kepler’s second law?
8) When you click on a planet, a display of information is show on the upper left. This include the planet’s distance from Sun and its sidereal period. According to K ...
1. NAME: __________________________________ Date: _______________
LAB: KEPLER’S LAWS OF PLANETARY MOTION
Purpose: To understand Kepler’s Laws describing the movements of planets in the solar system.
Background: In the 1500s, Nicolaus Copernicus challenged the GEOCENTRIC (earth-centered)
model of the solar system that had been promoted and accepted by philosophers and astronomers
such as Aristotle and Ptolemy for almost 2000 years. Copernicus described a HELIOCENTRIC
(sun-centered) model of the solar system, which placed Earth and the other planets in circular
orbits around the Sun. He proposed that all planets orbit in the same direction, but each planet
orbits at a different speed and distance from the Sun. Galileo Galilei’s observations made with
his telescope in the early 1600s and the work of other astronomers eventually confirmed
Copernicus’ model.
Tycho Brahe, a 16th
Century Danish astronomer, spent his life making detailed, precise
observations of the positions of stars and planets. His apprentice, Johannes Kepler, explained
Brahe’s observations in mathematical terms and developed three laws of planetary motion.
Kepler’s laws, together with Newton’s Laws of Inertia and Universal Gravitation, explain most
planetary motion.
KEPLER’S FIRST LAW
Kepler’s First Law, the “Law of Ellipses” states that all objects that orbit the Sun, including
planets, asteroids and comets, follow elliptical paths. An ellipse is an oval-shaped geometric
figure whose shape is determined by two points within the figure. Each point is called a
“focus” (plural: foci). In the solar system, the Sun is at one focus of the orbit of each planet; the
second focus is empty.
You will examine this by drawing 3 ellipses with foci that are different distances apart. (0.5 cm,
2 cm, and 4 cm)
Hypothesis:
Which ellipse will be most eccentric? ___________________________________
Why? ____________________________________________________________
_________________________________________________________________________
2. Procedure
1) If needed,tie the ends of the string into a loop about 6 - 7 cm across.
2) Fold the “drawings” paper in thirds, than flatten it out. The folds divide the paper into spaces
where you will draw and compare 3 ellipses. Measure and mark two dots 0.5 cm apart in the
center of the top third, mark two more 2 cm apart in the middle third, and in the bottom third,
make two dots 4 cm. apart. See Figure 2.
Figure 3
Figure 4
3) Put the paper over the cardboard, and push the thumbtacks into one set of points, far enough
to be firm, but not flat against the paper. These are the ellipses foci. Put the string around the
thumbtacks, and use the pencil inside it like a drawing pencil to draw an ellipse around the foci,
pulling the string tight against the tacks. See Figure 4. Have one partner hold the tacks steady if
needed.
4) Repeat step 3 for the other two sets of foci. It is OK if an ellipse goes off the paper at the
top and bottom, as long as the major axis (across the tacks) is on the paper. Put some scrap paper
down so you don’t draw on the desk : )
3. 5) Calculate Eccentricity (“out-of-roundness”)
ECCENTRICITY is the amount of flattening of an ellipse, or how much the shape of the ellipse
deviates from a perfect circle. A circle, which has only one central focus) has an eccentricity of
0. The greater the eccentricity, the less circular the ellipse.
a) Measure the distance between the thumbtacks for each ellipse. Put the data in the chart
below. (a)
b) Draw a line across the foci to the edges of the ellipse. This is the major axis. Measure
that. (b)
c) Calculate the eccentricity. (c) Eccentricity (e) = F ÷ A It should be between 0 and 1.
SHOW WORK IN THE DATA TABLE!
Why does elliptical eccentricity have no unit?
_________________________________________________________________________
Ellipse (a) Distance
between foci
(mm)
(b) Length
of major
axis (mm)
(c) Eccentricity: a/b
SHOW WORK!
Which is
roundest?
Least round?
A
B
C
6) Look at the Planet Data Table in your notes.
Which planet has the most elliptical orbit? ______________________________
Which planet has the most circular orbit? ______________________________
4. 7) Because a planet’s orbit is elliptical, its distance from the Sun varies throughout its
“year” (one revolution around the Sun). Look up the following terms in your textbook (p. 29
of blue textbooks in the room or p.668 yellow textbook) and write their definitions in the
spaces provided.
PERIHELION
APHELION
LABEL the points that represent perihelion and aphelion on the ellipse diagram on the
front of this page.
8) Look up the following information in your notes/textbook and fill in the blanks.
Earth is at perihelion on _________________________; on that date, Earth is
approximately _________________________ km from the Sun.
Earth is at aphelion on _________________________; on that date, Earth is
approximately _________________________ km from the Sun.
9) Based on your understanding of Kepler’s First Law, explain why the distance from a planet to
the Sun is typically given as an average distance.
5. KEPLER’S SECOND LAW
Kepler’s Second Law, the “Law of Equal Areas” states that a line drawn from the Sun to a planet
sweeps equal areas in equal time, as illustrated on the diagram on the next page. A planet’s
orbital velocity (the speed at which it travels around the Sun) changes as its position in its orbit
changes. Its velocity is fastest when it is closest to the Sun and slowest when it is farthest from
the sun.
http://library.thinkquest.org/03oct/02144/pics/basics/kepler2.png
1) If Area X = Area Y on the diagram above, what can be inferred about the orbital velocities as
the planet travels along its orbit through Area X compared to Area Y? (Which is faster?)
_____________________________________________________________________
_____________________________________________________________________
2) A planet’s orbital velocity is fastest at the position it its orbit called __________________
(perihelion/aphelion). Look back to your notes for the date when Earth is at this position.
During what season (in the Northern Hemisphere) is Earth at this position?
___________________________
Therefore, Earth moves ________________________ (faster/slower) in summer than in
winter, so summer in the Northern Hemisphere must be ___________________________
(longer/shorter) than winter.
3) Isaac Newton later determined that the force of GRAVITY holds the planets in orbit around
the Sun. When a planet is closer to the Sun, the force of the Sun’s gravitational attraction
on the planet is _________________________ (stronger/weaker) than when the planet is
farther from the Sun.
6. KEPLER’S THIRD LAW
Kepler’s Third Law, the “Law of Periods” relates a planet’s period of revolution (the time it takes
to complete one orbit of the Sun) to its average distance from the Sun. Kepler determined the
mathematical relationship between period and distance and concluded that the square of a
planet’s period is proportional to the cube of its mean distance from the Sun. The formula used
to determine this relationship for any planet is: T2
= R3
, where T is the planet’s period in Earth
years and R is the planet’s mean distance from the Sun in astronomical units (AU, where 1 AU
equals the mean distance from the Earth to the Sun = 150 million km).
Sample Problem: Planet X has an average distance from the Sun of 1.76 AU.
What is the planet’s period of revolution, in Earth years?
T2
= R3
T2
= (1.76)3
= 5.45
T =√ 5.45 = 2.33 Earth years
1) Calculate the period of revolution of each of the following
planets.
Planet Mean Distance to Sun (AU) Period of Revolution (Earth years)
Mercury 0.387
Mars 1.524
Saturn 9.539
Pluto 39.440
2) Haley’s comet has an average distance of 17.91 AU from the Sun. Calculate the period of
Haley’s comet. SHOW YOUR WORK BELOW!
7. 3) Draw a graph that shows the relationship between a planet’s period of revolution in Earth
years (some planets will need to be converted from days to years) and its average distance
from the Sun (in AU). Look up the data on the Planet Data Table in your notes. Plot period
on the x-axis and distance on the y-axis. Label each planet on the graph. Be sure to label
the axes and include a title.
Describe the graph. What is the relationship between period and distance from the Sun?