Published on Astronomy 801: Planets, Stars, Galaxies, and the Universe (https://www.e-education.psu.edu/astro801)

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Lesson 2

Overview

About Lesson 2

Like many students, you may have come into a course on astronomy thinking that we would spend an entire semester on night sky observations. What we really want to study, though, is astrophysics—we want to understand how those objects that you can observe behave and why they behave the way they do. Traditionally, this is taught from a historical perspective. We will see how over long periods of time we went from making observations of the objects in the sky to the first understanding of those objects.

In this lesson, we are going to begin studying the fundamental physics that is the foundation of astronomy; for now, we will focus on the orbits of the planets around the Sun and the force of gravity. The story involves many of the most famous scientists from throughout history: Aristotle, Ptolemy, Galileo, Copernicus, Newton, and some famous astronomers that you may not be as familiar with—Tycho Brahe and Johannes Kepler. The story of how our understanding of the solar system and the Earth’s place in it evolved is an excellent example of the process of science and how accurate observations can force us to change some of our most fundamental theories about the universe.

What will we learn in Lesson 2?

By the end of Lesson 2, you should be able to:

  • interpret the observational evidence for a heliocentric Solar System;
  • quantitatively compare and contrast the shape of the planetary orbits and the relationships between their distances from the Sun and their orbital periods;
  • explain how an orbit is a balance between the force of gravity and the tangential motion of an object;
  • describe how the orbital properties of an object can be used to determine the mass of the system.

What is due for Lesson 2?

Lesson 2 will take us one week to complete. Please refer to the Calendar in Canvas for specific time frames and due dates.

There are a number of required activities in this lesson. The following table provides an overview of those activities that must be submitted for Lesson 2.

Lesson 2 Activities
Requirement Submitting Your Work
Lesson 2 Quiz Your score on this Canvas quiz will count towards your overall quiz average.
Lesson 2 Practice Math Problems There is a second quiz for this lesson in the Lesson 2 Module in Canvas. This one is all short math problems. You will be graded only on effort on this quiz, that is you will be graded for taking it and working on the problems, but not on your answers.
Lab 1 During Lesson 2, you should begin taking data for the "Moons of Jupiter" lab you will complete at the end of Lesson 3. You do not need to submit anything this week.

Questions?

If you have any questions, please post them to Piazza (not email). I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

Naked Eye Observations

Additional reading from www.astronomynotes.com [1]

  • Planetary motions [2]

Prior to the invention of the telescope, an observer could see the following objects with the unaided eye:

  • the Sun;
  • the Moon;
  • the five planets: Mercury, Venus, Mars, Jupiter, and Saturn;
  • the stars.

If you recall from Lesson 1, we discussed that the position of the Sun in the sky appears to drift with respect to the background stars (we didn’t discuss it in depth, but the Moon also drifts with respect to the stars–this should be more obvious because we can observe it night after night!). We can’t see the stars during the day, so the Sun’s drift is not obvious to most of us. However, we know that many civilizations in the pre-telescopic era were familiar with the drift of the Sun with respect to the stars. For example, they carefully studied the heliacal risings and settings of stars and used these to mark dates on their calendars. The heliacal rising of a star is the first day it is visible just before dawn, which is a direct indication of the Sun’s drift with respect to the stars. If the Sun happens to be from our point of view in front of a particular star, say Sirius, that star will rise just before dawn only on one day of the year. The next day, Sirius will rise four minutes earlier because of the Sun’s eastward drift along the ecliptic.

Check this out...

There is more information on helical risings from Stanford on their site: "Ancient Observatories, Timeless Knowledge [3]".

As you can see from the list in the first paragraph, there are only five planets visible to the naked eye. For these same observers, what distinguished planets from stars is, again, their motion. Planets also appear to drift compared to the background stars but in a more complicated manner than the Sun.

Test this with Starry Night!

Let's investigate two examples...

  1. Open up Starry Night. Set it to the following:
    • location = somewhere in Pennsylvania
    • date & time = May 6, 2008 @ 9:30pm
    • time flow rate = 1 sidereal day
    • constellation labels and stick figures turned on (if you press "k" on the keyboard, it accomplishes this for you without having to use the menu or options tab)
  2. Press play (or click on the forward one step button at least a dozen times), and you will watch Mars' location change with respect to the stars over the course of a few weeks or a few months.
  3. For the second example, open a new Starry Night session and set it to the following:
    • date & time = June 6, 2003 @ 3:30am
    • Otherwise, use all of the same settings as you used above.
  4. Again, press play or click on the forward one step button at least a dozen times. How did the path of Mars differ between the two examples?
Some of the more expensive versions of Starry Night (Pro and Pro Plus) allow you to turn on the "local path," which just plots a curve showing you the path of an object on the sky. I created a saved file using Starry Night Pro Plus, and if you open this document in your version of Starry Night, it will allow you to see the local path of Mars, which makes the phenomenon I want you to observe more obvious. Copies of the save files for these examples are posted for you in Canvas or can be downloaded here:  May 2008 [4] and June 2003 [5].

Below is a composite image created of Mars during the 2003 time period you simulated with Starry Night:

Composite photo image of path of Mars compared to the stars over many nights to illustrate a "retrograde loop."
Credit: Astronomy Picture of the Day [6]

You may want to try to avoid reading the caption at this link, though, because it gives away the punchline to this lesson.

If you study your Starry Night path of Mars or the APOD image above, you will find that the first part of Mars’ motion is prograde, or eastward compared to the stars, just like the Sun. However, around July 30, Mars has slowed down and proceeds to move retrograde, or westward compared to the stars. Then, it slows down again and begins prograde motion again! If you study the other naked eye planets, for example, Jupiter and Saturn [7], you find that they exhibit the same behavior. This path is often referred to as a "retrograde loop." Again using Mars as an example, its retrograde loop is not always identical. The dates of the beginning and ending of the retrograde loop, the shape of the path of the loop with respect to the stars, and the point along the ecliptic at which the loop begins changes.

For the rest of this lesson, we are going to again study the geometry and motion of the solar system and, by the end, we will show that we can easily explain this complicated motion of Mars. While this seems simple in retrospect, it took thousands of years for scientists to determine the solution!

The Geocentric Model

Additional reading at www.astronomynotes.com [8]

  • Method for finding scientific truth [9]
    • A scientific theory is [10]
    • Ways of finding the truth [11]

Scientific Models

Before returning to retrograde Mars and beginning our discussion of the early attempts to explain this behavior, let's first discuss scientific models. This is terminology that is now being included in state science education standards and the Next Generation Science Standards (NGSS), and I want to be quite clear about what I mean when I use the term in this class.

To astronomers and other scientists, “making a model” has a specific meaning: taking into account our knowledge of the laws of science, we construct a mental picture of how something works. We then use this mental model to predict the behavior of the system in the future. If our observations of the real thing and our predictions from our model match, then we have some evidence that our model is a good one. If our observations of the real thing contradict the predictions of our model, then it teaches us that we need to revise our picture to better explain our observations. In many cases, the model is simply an idea—that is, there is no physical representation of it. So, if, when I use the word "model," you picture in your head a 1:200 scale copy of a battleship that you put together as a kid, that is not what is meant here. However, that doesn't preclude us from making a physical representation of the model. So, for example, if you are studying tornadoes, you can build a simulated tornado tube using 2 liter soda bottles filled with water. However, for it to be useful as a scientific model, you would want to use the physical model to try and study aspects of real tornadoes. In modern science, many models are computational in nature—that is, you can write a program that simulates the behavior of a real object or phenomenon, and if the predictions of your computer model match your observations of the real thing, it is a good computer model.

This is also a good time to introduce a statement referred to as Occam’s Razor. This is a simple statement that paraphrased says: If there are two competing models to explain a phenomenon, the simplest is the one most likely to be correct. This concept was taught to me in the following way: if you propose a model, you are only allowed to invoke the Easter Bunny once, but if you have to invoke the Easter Bunny twice (as in “then the Easter Bunny appears and makes this happen"), your model is probably wrong.

Want to learn more?

For more history, see a discussion of Occam's Razor on Wikipedia [12]. I realize that Wikipedia is not always to be considered a trusted resource, but this is a good overview.

Many textbooks use this example of the study of Mars as an opportunity to introduce the "scientific method".  In previous versions of this course, I did the same thing.  I learned, and I'm sure many of you learned, that the scientific method has 5 or so steps that, if done in order, you are correctly doing science.  However, even when I included that content in my course, I knew that I did not do science that way.  I finally changed this lesson in the course when a teacher I collaborated with said to me, "Do you ever do your science the way the scientific method is written about in textbooks?", and I said no.

What I hope will be made clear in the rest of the course is that in practice science is very non-linear.  In fact, as a fairly frequent judge for the "Pennsylvania Junior Academy of Science" (which may be similar to science fairs where you teach), I often complain about their rubric for judging, because they force students to try to approach science in a linear, step-by-step model.  Scientists all do the standard steps of the scientific method at some point, however, not necessarily in the order presented in textbooks or in a way that they identify as "Now I am on step 5 of the process", for example. This process is really completed by a community of scientists working on scientific problems separately.  Everyone involved in the process is working towards the same goal, but some may contribute observations while others build better models, for example. If you would like to discuss this more, this would be an excellent topic for Piazza!

The Greek's Geocentric model

Traditionally in Astronomy textbooks, the chapter on the topic of the motion of the planets in the sky almost always begins with mention of the ancient Greeks. I will not go into a lot of detail on the lives and accomplishments of Eratosthenes, Aristarchus, Hipparchus, etc., but I will follow tradition, and we will study here the model of the Universe presented by the Greeks. In particular, we will consider the work of Aristotle and Ptolemy, because their model was considered the best explanation for the workings of the solar system for more than 1000 years!

While I will gloss over most of the discoveries of the famous Greek philosophers (or mathematicians or astronomers, whatever you prefer to consider them), I think it is quite important to note that they were able to determine many sophisticated understandings of our Solar System based on their strong grasp of geometry. For example, Eratosthenes is given credit for demonstrating that the Earth is round and for performing the first experiment that resulted in a measurement of the circumference of the Earth [13].

Try This!

If you aren't familiar with Eratosthenes' experiment, I encourage you to spend time at the website above and to even consider repeating the experiment if you can find a partner located several hundred miles from your school.

Now, let's return to a discussion of the Greeks' model. Today, we start with our well known laws of physics as the basis of our scientific models. At the time that the Greek model was being developed, those laws were unknown, though, and instead they held firmly to several beliefs that formed the foundation of their model of the solar system. These are:

  • the Earth is the center of the universe and it is stationary;
  • the planets, the Sun, and the stars revolve around the Earth;
  • the circle and the sphere are “perfect” shapes, so all motions in the sky should follow circular paths, which can be attributed to objects being attached to spherical shells;
  • objects obeyed the rules of “natural motion,” which for the planets and the stars meant they orbited around the Earth at a uniform speed.

Given this set of rules (in modern scientific language, these would be referred to as the assumptions of the model; however, the Greeks believed these to be laws that could not be altered), the Greeks constructed a model to predict the positions of the planets. They knew about retrograde motions, and, therefore, they also constructed their model in such a way to account for the retrograde motions of the planets. Their model is referred to as the geocentric model because of the Earth’s place at the center.

Our knowledge of the Greek’s Geocentric model comes mostly from the Almagest, which is a book written by Claudius Ptolemy about 500 years after Aristotle’s lifetime. In the Almagest, Ptolemy included tables with the positions of the planets as predicted by his model. If you recall from our previous discussion, the retrograde motions of the planets are very complex; therefore, Ptolemy had to create an equally complex model in order to reproduce these motions. I will quickly summarize things here: Ptolemy’s model did not simply have the planets and the Sun attached to one sphere each, but he had to adopt circles (epicycles) on top of circles (deferents) with the Earth offset from the center. The most complex version of the model was still often in error in its predictions by several degrees, or by an angular distance larger than the diameter of the full Moon.

Want to learn more?

This is an interesting topic I won't describe in any more detail, but if you would like to learn more, there is much more about the Ptolemaic model in most introductory astronomy textbooks, including the online Astronomynotes.com [14].

There is a faculty member at Florida State who has made Flash models of the Ptolemaic system. For example, you can see how the Moon and Sun or Mercury and the Sun were supposed to have orbited Earth:

  • The final Almagest Moon [15]
    • (Try this version of the final Almagest Moon [16] if the model at that link does not work in your browser.)
  • Mercury [17]
    • (Try this version of Mercury [17] if the model at that link does not work in your browser.)

Recall that the Greeks did rely on mathematical reasoning when conducting experiments and designing their models. You may wonder, in the Greek model, what order were the "planets" out from the Earth, and how were they chosen to be in that order? The order was:

  1. Earth (unmoving; located at the center)
  2. Moon
  3. Mercury
  4. Venus
  5. Sun
  6. Mars
  7. Jupiter
  8. Saturn

We will discuss this concept more later, but consider the angular speed of an object on the sky. The faster the angular speed, the larger the angular distance an object will cover in the same amount of time. A simple example is to consider two airplanes on the sky. One is close to you, and the other more distant. If both planes are flying at the same speed in the same direction across your line of sight, the more distant airplane will appear to cover a shorter angular distance on the sky than the nearby plane. So, if you can estimate the angular speed of two objects and if you assume that they are moving at the same real speed and in the same direction, the one that travels the shorter distance on the sky must be the more distant object.

The Greeks used this method to estimate the distance to the planets, and they were able to determine the relative ordering of the planets. The most significant flaw was their assumption of the Earth as the center of all things.

The Heliocentric Model

Additional reading at www.astronomynotes.com [1]

  • Copernicus' heliocentric universe [18]

The geocentric model of the Solar System remained dominant for centuries. However, because even in its most complex form it still produced errors in its predictions of the positions of the planets in the sky, some astronomers continued to search for a better model.

The astronomer given the credit for presenting the first version of our modern view of the Solar System is Nicolaus Copernicus, who was an advocate for the heliocentric, or Sun-centered model of the solar system. Copernicus proposed that the Sun was the center of the Solar System, with all of the planets known at that time orbiting the Sun, not the Earth. Although this solved many longstanding problems in the Ptolemaic model, Copernicus still believed that the orbits of planets must be circular, and so his model was not much more successful than Ptolemy’s in predicting the position of the planets. His model was very successful, however, in solving the problem of retrograde motion in a very elegant manner. This is illustrated in the animation below. Click on the "start" button to see the retrograde motion.

Animation Caption:

This animation created with Starry Night begins with a top-down view of the Solar System with the orbital paths of the inner planets shown. It zooms out to show only the Earth and Mars. Lines are drawn from Earth to Mars to show where on the sky someone on Earth looking at Mars would see Mars with respect to the background stars. Each arrow is labeled with the date on Earth, and as both planets orbit the Sun, we see that the position of Mars with respect to the stars appears to move as time goes forward. In May of 2016, the Earth overtakes Mars, so the arrow for this date appears to move in the opposite direction compared to the previous dates. This continues for June and July before Mars appears to move in the original direction again beginning in August.

The solution to the problem of retrograde motion is to realize that the Earth is moving more quickly around the Sun than Mars. Along its orbit, Earth will at some times lag behind Mars from an angular point of view. That is, if Earth is at the 3 o’clock position along its orbit, Mars may be at 1 o’clock. Since Earth moves faster along its path, Earth will overtake Mars as they both hit the 12 o’clock position at the same time. After passing Mars, Earth will reach the 9 o’clock position on its orbit while Mars only makes it to 11 o’clock. From our point of view on Earth, Mars will appear to move prograde on the sky when we are approaching it; however, as we overtake Mars (which you can see in the animation if you replay it and watch the relative positions of the two planets closely), it will appear to come to a stop and then begin to move retrograde. A good analogy to help clarify this concept is to visualize runners on a track. Imagine two runners, one moving quickly in an inside lane (Earth) and another moving more slowly on the outside lane (Mars). When both are on the straightaways, the Earth runner will see the Mars runner moving forward but slowing down as the Earth runner catches up. However, when both hit the turn, the Earth runner will pass Mars, who will seem to be moving backwards (or retrograde!) from Earth's point of view.

Test this with Starry Night!

Note also that you can reproduce the animation (but without the arrows) with Starry Night! This is a bit more tricky, but here are the steps:

  1. Instead of choosing a location on Earth or on Mars, you can choose a stationary location. In this case, you want to be floating above the Sun, so you can set the location to X = 0, Y = 0, and Z = 1 billion miles (or in Astronomical Units, 10 AU).
  2. Choose to label planets and moons from the labels menu
  3. If you do not see the Sun and planets, search for the Sun in the find menu and double click on the word "Sun" when it comes up
  4. Right click on Earth and Mars and choose "orbit"
  5. Set the time step to days; press play

You can now watch the orbits of Earth and Mars on a given set of dates to choose when Earth is overtaking Mars, and then you can reset things so you are watching the sky from Earth on that same date and watch Mars go through a retrograde loop! I have not created a Starry Night file for this example, but please let me know if you would like one.

Starry Night does have some built in "Favorites".  They do have a similar one for the inner Solar System. In the Favorites menu, choose Solar System, then Inner Planets, and then Inner Solar System, and it will show you a view of the Inner Solar System slightly different from the one you will see if you follow the instructions above. You can also get to this Favorite by clicking on the "hamburger menu" (the three horizontal lines) on the right side of the top status bar.

Although Copernicus’ model solved some problems, its lack of accuracy in predicting planetary positions kept it from becoming widely accepted as better than the Ptolemaic model. The advocates for the Geocentric model also proposed another test for the heliocentric model: if the Earth is orbiting the Sun, then the distant stars should appear to shift from our point of view, an effect known as parallax. We will study parallax in more detail in a later lesson on stars. However, for now I will note that this caused a problem for advocates of the heliocentric model. If they were right, we should observe parallax, but not even the most accurate observers of the day were able to detect a measurable amount of parallax for even a single star.

Forgetting parallax for a moment, the advances necessary to increase the acceptance of the heliocentric model came from Tycho Brahe and Johannes Kepler. Brahe is credited with being one of the best observers of his time. At his observatory, and over approximately 15 years, using instruments he designed and built, Brahe compiled a continuous list of accurate positions for the planets on the sky. Johannes Kepler came to work with Brahe shortly before Brahe died. Kepler used his mathematical skill to study the accurate observations of Brahe and then proposed three laws that accurately describe the motions of the planets in the solar system.

Kepler's Three Laws

Additional reading at www.astronomynotes.com [8]

  • Kepler's laws of planetary motion [19]

Diagram showing an exaggerated elliptical path of a planet orbiting the sun.

Elliptical path of planet orbiting the Sun
Credit: Wikimedia Commons [20]

First Law

Kepler was a sophisticated mathematician, and so the advance that he made in the study of the motion of the planets was to introduce a mathematical foundation for the heliocentric model of the solar system. Where Ptolemy and Copernicus relied on assumptions, such as that the circle is a “perfect” shape and all orbits must be circular, Kepler showed that mathematically a circular orbit could not match the data for Mars, but that an elliptical orbit did match the data! We now refer to the following statement as Kepler’s First Law:

  • The planets orbit the Sun in ellipses with the Sun at one focus (the other focus is empty).

For more information about ellipses, you can read in gory mathematical detail the page hosted at Mathworld [21], and there is also information on ellipses in Wikipedia [22].

Here is a demonstration of the classic method for drawing an ellipse:

Photo of someone following the classic method for drawing an ellipse using a loop of string around two tacks separated by a small distance.
Drawing an ellipse
Credit: Wikipedia [23]

The two thumbtacks in the image represent the two foci of the ellipse, and the string ensures that the sum of the distances from the two foci (the tacks) to the pencil is a constant. Below is another image of an ellipse with the major axis and minor axis defined:

Diagram of a drawing of an ellipse, showing definition of major and minor axes and foci.
Diagram of an ellipse
Credit: Wikipedia [24]

We know that in a circle, all lines that pass through the center (diameters) are exactly equal in length. However, in an ellipse, lines that you draw through the center vary in length. The line that passes from one end to the other and includes both foci is called the major axis, and this is the longest distance between two points on the ellipse. The line that is perpendicular to the major axis at its center is called the minor axis, and it is the shortest distance between two points on the ellipse.

In the image above, the green dots are the foci (equivalent to the tacks in the photo above). The larger the distance between the foci, the larger the eccentricity of the ellipse. In the limiting case where the foci are on top of each other (an eccentricity of 0), the figure is actually a circle. So you can think of a circle as an ellipse of eccentricity 0. Studies have shown that astronomy textbooks introduce a misconception by showing the planets' orbits as highly eccentric in an effort to be sure to drive home the point that they are ellipses and not circles. In reality the orbits of most planets in our Solar System are very close to circular, with eccentricities of near 0 (e.g., the eccentricity of Earth's orbit is 0.0167). For an animation showing orbits with varying eccentricities, see the eccentricity diagram [25] at "Windows to the Universe." Note that the orbit with an eccentricity of 0.2, which appears nearly circular, is similar to Mercury's, which has the largest eccentricity of any planet in the Solar System. The elliptical orbits diagram [26] at "Windows to the Universe" includes an image with a direct comparison of the eccentricities of several planets, an asteroid, and a comet. Note that if you follow the Starry Night instructions on the previous page to observe the orbits of Earth and Mars from above, you can also see the shapes of these orbits and how circular they appear.

Kepler’s first law has several implications. These are:

  • The distance between a planet and the Sun changes as the planet moves along its orbit.
  • The Sun is offset from the center of the planet’s orbit.

Second Law

In their models of the Solar System, the Greeks held to the Aristotelian belief that objects in the sky moved at a constant speed in circles because that is their “natural motion.” However, Kepler’s second law (sometimes referred to as the Law of Equal Areas), can be used to show that the velocity of a planet changes as it moves along its orbit!

Kepler’s second law is:

  • The line joining the Sun and a planet sweeps through equal areas in an equal amount of time.

The image below links to an animation that demonstrates that when a planet is near aphelion (the point furthest from the Sun, labeled with a B on the screen grab below) the line drawn between the Sun and the planet traces out a long, skinny sector between points A and B. When the planet is close to perihelion (the point closest to the Sun, labeled with a C on the screen grab below), the line drawn between the Sun and the planet traces out a shorter, fatter sector between points C and D. These slices that alternate gray and blue were drawn in such a way that the area inside each sector is the same. That is, the sector between C and D on the right contains the same amount of area as the sector between A and B on the left.

Graphic illustration showing how changes in a planet's velocity as it moves along its orbit is equivalent to sweeping out equal area in equal times. [27]
Click on this image to launch the animation in Windows Media Player.  It shows a planet sweeping out equal areas in equal times.
Kepler's 2nd Law
Credit: Dr. Michael Gallis, Penn State Schuylkill

Since the areas of these two sectors are identical, then Kepler's second law says that the time it takes the planet to travel between A and B and also between C and D must be the same. If you look at the distance along the ellipse between A and B, it is shorter than the distance between C and D. Since velocity is distance divided by time, and since the distance between A and B is shorter than the distance between C and D, when you divide those distances by the same amount of time you find that:

  • A planet is moving faster near perihelion and slower near aphelion.

The orbits of most planets are almost circular, with eccentricities near 0. In this case, the changes in their speed are not too large over the course of their orbit.

For those of you who teach physics, you might note that really, Kepler's second law is just another way of stating that angular momentum is conserved. That is, when the planet is near perihelion, the distance between the Sun and the planet is smaller, so it must increase its tangential velocity to conserve angular momentum, and similarly, when it is near aphelion when their separation is larger, its tangential velocity must decrease so that the total orbital angular momentum is the same as it was at perihelion.

Third Law

Kepler had all of Tycho’s data on the planets, so he was able to determine how long each planet took to complete one orbit around the Sun. This is usually referred to as the period of an orbit. Kepler noted that the closer a planet was to the Sun, the faster it orbited the Sun. He was the first scientist to study the planets from the perspective that the Sun influenced their orbits. That is, unlike Ptolemy and Copernicus, who both assumed that the planet's “natural motion” was to move at constant speeds along circular paths, Kepler believed that the Sun exerted some kind of force on the planets to push them along their orbits, and because of this, the closer they are to the Sun, the faster they should move.

Kepler studied the periods of the planets and their distance from the Sun, and proved the following mathematical relationship, which is Kepler’s Third Law:

  • The square of the period of a planet’s orbit (P) is directly proportional to the cube of the semimajor axis (a) of its elliptical path.
  • P 2  ∝  a 3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

What this means mathematically is that if the square of the period of an object doubles, then the cube of its semimajor axis must also double. The proportionality sign in the above equation means that:

  • P 2  = k a 3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

where k is a constant number. If we divide both sides of the equation by a 3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , we see that:

  • P 2  /  a 3  = k This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

This means that for every planet in our solar system, the ratio of their period squared to their semimajor axis cubed is the same constant value, so this means that:

  • ( P 2  /  a 3 )   Earth  = ( P 2  /  a 3 )   Mars  = ( P 2  /  a 3 )   Jupiter This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

We know that the period of the Earth is 1 year. At the time of Kepler, they did not know the distances to the planets, but we can just assign the semimajor axis of the Earth to a unit we call the Astronomical Unit (AU). That is, without knowing how big an AU is, we just set a Earth  = 1 AU This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . If you plug 1 year and 1 AU into the equation above, you see that:

  • ( P 2  /  a 3 )   Earth  = ( P 2  /  a 3 )   Mars  = ( P 2  /  a 3 )   Jupiter This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So for every planet, P 2  /  a 3  = 1  This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. if P is expressed in years and a is expressed in AU. So if you want to calculate how far Saturn is from the Sun in AU, all you need to know is its period. For Saturn, this is approximately 29 years. So:

  • ( P 2  /  a 3 )   Saturn  =  ( 29 years ) 2  /  ( a AU ) 3  = 1 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
  • ( a AU ) 3  = 841 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
  • (a AU) =  3 √ 841 = 9.4 AU This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So Saturn is 9.4 times further from the Sun than the Earth is from the Sun!

Newtonian Gravitation

Additional reading at www.astronomynotes.com [1]

  • Newton's laws of motion [28]
  • Universal law of gravity [29]

Kepler's Laws are sometimes referred to as "Kepler's Empirical Laws." The reason for this is that Kepler was able to mathematically show that the positions of the planets in the sky were fit by a model that required orbits to be elliptical, the velocity of the planets in orbit to vary, and that there is a mathematical relationship between the period and the semimajor axis of the orbits. Although these were remarkable accomplishments, Kepler was unable to come up with an explanation for why his laws were true—that is, why are orbits elliptical and not circular? Why does the period of a planet determine the length of its semimajor axis?

Isaac Newton is given credit for explaining, theoretically, the answers to these questions. In his most famous work, the Principia, Newton presented his three laws:

  1. An object at rest or in motion in a straight line at a constant speed will remain in that state unless acted upon by a force.
  2. The acceleration of a body due to a force will be in the same direction as the force, with a magnitude indirectly proportional to its mass. (This is usually written as F = ma, or Force = mass x acceleration This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ).
  3. For every action, there is an equal and opposite reaction.

In addition, he presented his law of universal gravitation:

  • The force of gravity between two masses is: F = G ( m 1  x  m 2 ) /  d 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

That is, the force of gravity depends on both their masses, a constant (G), and it drops off as 1 over the distance squared. In this equation, d, the distance, is measured from the center of the object. That is, if you want to know the force of gravity on you from the Earth, you should use the radius of the Earth as d, since you are that far away from the center of the Earth.

Using these laws and the mathematical techniques of calculus (which Newton invented), Newton was able to prove that the planets orbit the Sun because of the gravitational pull they are feeling from the Sun. The way an orbit works is as follows (this is a thought experiment attributed to Newton, sometimes called Newton's cannon):

Think of a cannon on a high mountain near the north pole of the Earth. If you were to shoot a cannonball horizontally, parallel to the Earth's surface, it would drop vertically towards the Earth's surface at the same time it is moving horizontally away from the mountain, and eventually hit the Earth. If you shot the cannonball with more force, it would travel farther from the mountain before it hit the Earth. Well, what would happen if you shot the cannonball with so much force that the amount of the vertical drop of the cannonball towards the surface due to Earth's gravity was the same magnitude as the Earth's dropoff because of its spherical shape? That is, if you could shoot a projectile with enough force, it would fall towards the Earth like any other projectile, but it would always miss hitting the Earth! For an example of this, see this Applet of Newton's cannon [30].

Although the Earth was never shot out of a cannon, the same physics applies. Think of the Earth sitting at the 3 o'clock position in its orbit around the Sun. If the Earth were to just freely fall through space without experiencing any force, by Newton's first law, it would just continue to fall in a straight line. However, the Sun is pulling on the Earth such that the Earth feels a tug towards the Sun. This causes the Earth to also fall towards the Sun a bit. The combination of the Earth falling through space and it perpetually being tugged a little bit in the direction of the Sun causes it to follow a roughly circular path around the Sun. This effect can be illustrated in the following animation:

Animation caption

This animation shows the Earth at one point along its orbit around the Sun. It labels the tangential velocity of the Earth with an arrow that is tangential to Earth's orbit. It also labels the force of gravity pulling Earth towards the Sun, which is perpendicular to the tangential velocity of the Earth. The combination of these two factors causes the Earth to accelerate and follow an elliptical path around the Sun.

I should note here that this concept requires thinking about the concept of inertia, which can be very confusing, and, in fact, this particular animation uses terminology that may reinforce this confusion. There is no "Force of Inertia"; inertia is not a force. Instead, the proper way to think about this is that inertia is the property of an object that determines how strongly it resists changing its motion. So, picture a planet moving in a straight line; it has a lot of inertia, because a large, massive planet is hard to move off of that straight line. However, the Sun pulls on that planet with the force of gravity, and that gravitational pull is strong enough to divert the planet from a straight line path. If the tangential velocity of the planet is balanced by the change in velocity introduced by gravity, you get a stable orbit.

Try this!

The PHeT simulations [31] include two that allow you to play with orbital motion. One is called "Gravity and Orbits" and the other is called "My Solar System". You can set up initial conditions for planets and a star, and then see what happens.

  1. Can you create a stable orbit?
  2. Can you alter a stable orbit so that it is no longer stable? If so, what did you do?
  3. What happens to objects on unstable orbits?

Using the techniques of calculus, you can actually derive all of Kepler's Laws from Newton's Laws. That is, you can prove that the shape of an orbit caused by the force of gravity should be an ellipse. You can show that the velocity of an object increases near perihelion and decreases near aphelion, and you can show that P 2  = k a 3 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . In fact, Newton was able to derive the value for the constant, k, and today we write Newton's version of Kepler's Third Law this way:

P 2  = ( 4π   2  x  a 3 ) / G( m 1  +  m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

Which means that k = 4π   2  / G(m 1  + m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

If we use Newton's version of Kepler's Third Law, we can see that if you can measure P and measure a for an object in orbit, then you can calculate the sum of the mass of the two objects! For example, in the case of the Sun and the Earth, m 1  =  m Earth , and  m 2 =  m Sun This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , so just by measuring P Earth  and  a Earth This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , you can calculate m Sun  +  m Earth This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. !

This is the basis of a lab we are going to do during this unit. You are going to find P and a for several of Jupiter's Moons, and you are going to use those data to calculate the mass of Jupiter.

Lastly, I would like everyone to do a quick calculation using the formula for Newton's Law of Universal Gravitation:

F = G ( m 1  x  m 2 ) /  d 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

For now, we can ignore the constant G. We are going to calculate a ratio, so in the end the constant will drop out. What I want us to look at is the force of gravity "in space." That is, for astronauts in the space shuttle or in the International Space Station (ISS), how does the force of gravity from Earth that they feel compare to the force of gravity that you feel sitting here on Earth?

If you are unfamiliar with doing ratios, do the following step by step:

  1. Write out this equation one time for the situation on Earth, that is:

    F onEarth  = G ( m 1  x  m 2 ) /  d onEarth 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
  2. Write out this equation a second time for the situation in Space, that is:

    F inSpace  = G ( m 1  x  m 2 ) /  d inSpace 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
  3. Form a ratio taking the equation from #1 above and putting it over #2 above, that is:

    ( F onEarth ) / ( F inSpace ) = ( G ( m 1  x  m 2 ) /  d onEarth 2 ) / ( G ( m 1  x  m 2 ) /  d inSpace 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

At this point, if you recall from the rules of algebra, when you have quantities on the top and bottom of a fraction that are the same, they cancel out. So, you can cross out everything on the right hand side you find on both the top and bottom, that is, G, m1, and m2.

You are then left with:

( F onEarth ) / ( F inSpace ) = ( 1 /  d onEarth 2 ) / ( 1 /  d inSpace 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

What this tells you is that the ratio between the force of gravity you feel on Earth to the force of gravity you feel in space is only related to the distance between Earth and you in both cases. In case 1, when you are on Earth, you would fill in the radius of the Earth, approximately 6400 km. The space shuttle and the ISS do not orbit far from Earth. A reasonable number for the distance between the surface of Earth and the ISS is about 350 km. So, the distance between the Earth and the ISS for calculating the force of gravity on the ISS is ( 6400 km + 350 km ) = 6750 km This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . Fill in these values for d onEarth  and  d inSpace   This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. and calculate this ratio. This will give you an answer for how much stronger the gravity is on the surface of Earth compared to in the ISS.

Tell us about it!

What is the answer? Does it surprise you? Is there gravity in space? Why are astronauts in the ISS "weightless"? Record your thoughts in the Comments area.

Additional Resources

  1. I tend not to focus too much on the historical figures in astronomy and their accomplishments, and I left out a lot about Galileo, who was very important in the transition from a Geocentric to Heliocentric view of the Universe. If you'd like to read more about Galileo on your own, I recommend the Galileo Project. [32]
  2. In the ClassAction Modules [33] there are modules related to this unit under Basic Motions & Ancient Astronomy and Renaissance Astronomy. I recommend the Renaissance Astronomy module called the "Planetary Orbit Simulator," and they also have modules similar to the ones in this lesson (e.g., the retrograde motion animation).
  3. On Teacher's Domain, there are several resources related to this lesson:
    • A video from NASA [34] of the Galileo Experiment performed on the Moon.
    • A video about a classroom demo you can do on weightlessness [35].
    • A NOVA animation about Newton's cannon and weightlessness [36].
  4. In the lesson, I recommended the PhET simulations [37] and suggested you try experimenting to create stable and unstable orbits using one of their orbital simulators. A physics teacher collaborator and I wrote an article about an in-class investigation you can do with one of those simulations, and it is available for free in PDF form here: Earth Scientist Vol XXIX Summer 2013 [38] (see p. 32)

Tell us about it!

Have another website on this topic that you have found useful? Share it in the Comment area!

 

Summary & Final Tasks

In this lesson, you learned how astronomy went from a careful study of the sky to a science with a basis in the fundamental laws of physics. While in an astronomy course, we are forced to gloss over a lot of the physics of Newton's Laws, I do hope that you come away with an appreciation for the physics of orbits and a firm understanding that there is gravity in space.

Activity 1

Directions

First, please take the Web-based Lesson 2 quiz.

  1. Go to Canvas.
  2. Go to the "Lesson 2 Quiz" and complete the quiz.

Good luck!


Activity 2

Directions

There is a second quiz for this lesson that contains several short math problems. While I expect you to complete the quiz and to give it the same effort you would for a graded assignment, for these problems you will only be graded on completion and not on the accuracy of your answers. Your participation in this quiz will count in lieu of a discussion forum for this lesson.

  1. Go to Canvas.
  2. Go to the "Lesson 2 Practice Problems" and complete the quiz.

Activity 3

Directions

During this week, you should begin work on the lab exercise that will be completed and submitted next week.

  1. Lab 1 is located on the next page.
  2. Read the background information, and then follow the instructions for taking data with the simulated Moons of Jupiter.
  3. Take data and record it as instructed.

Reminder - Complete all of the lesson tasks!

You have finished the reading for Lesson 2. Double-check the list of requirements on the Lesson 2 Overview page to make sure you have completed all of the activities listed there before beginning the next lesson.

Tell Us About It!

If there is anything you'd like to comment on, or add to, the lesson materials, feel free to post your thoughts in the Comment area. For example, what did you have the most trouble with in this lesson? Was there anything useful here that you'd like to try in your own classroom?

Don't see the "Comment" area? You need to be logged in to this site first! Once you have logged in, you may need to refresh the page in order to see the comment area.

 

Lab 1, Part 1

Background

One of the observations that Galileo is famous for making is the discovery of four Moons of Jupiter [39], which these days we refer to as the "Galilean Moons."

This was one of the observations that contributed to the revolution in our understanding of the true nature of the Solar System. What Galileo personally observed was what he thought were stars near Jupiter, and night after night, he witnessed their positions change with respect to Jupiter.

Test this with Starry Night!

I have created a Starry Night save file (.snf) [40] to let you jump to see this directly (a copy is posted in Canvas). If you would like to set it up yourself, you can do the following:

  • set the date and time to June 28, 2009 at 2:00am;
  • turn on the Planet / Moon labels so you can see which objects are visible;
  • either by mousing around or using the Find feature, locate Jupiter and right-click on it to center it;
  • zoom in so the field of view is approximately 40 arcminutes wide;
  • set the time flow rate to 1 day;
  • click on the forward one step button to watch the positions of the objects change as each day goes by.

You should witness exactly what Galileo did—as you click on the forward button, each night the arrangement of the four Galilean Moons (Io, Europa, Ganymede, and Callisto) changes with respect to Jupiter.

Now, let's do something that Galileo could not. Let's look at Jupiter from above its North Pole so that we can see physically what is going on.

Test this with Starry Night!

I have created a Starry Night save file (.snf) [41] to let you jump to see this directly (a copy is posted in Canvas), but if you would like to set it up yourself you can do the following:

  • set the viewing location to the surface of Jupiter, and set the latitude and longitude to 90 degrees North, 0 degrees West;
  • turn on the Planet / Moon labels so you can see which objects are visible;
  • use the up / down arrows to move about 0.01 AU or so above Jupiter's North Pole;
  • set the time flow rate to 1 hour;
  • click on the forward one step button to watch the positions of the objects change as each hour goes by.

What you can see in this latter view is the orbit of the moons, but what you see in the former view is what appears to be a side to side change in position of the moons. In this lab, we are going to measure that side to side motion and use that data to calculate the mass of Jupiter using Newton's version of Kepler's Third Law.

Let's talk about how the side view and top view compare. Below is an image that shows the top view (that is, as seen from Jupiter's North Pole) of the orbit of a moon:

Graphic illustration to show the overhead view (above Jupiter's N. pole) of the orbit of a Galilean moon.
Image of a hypothetical moon orbiting a hypothetical planet with the radius labeled and the apparent separation as seen by a distant observer labeled
Credit: Penn State Astronomy & Astrophysics

If you study the image, you will note that when the moon is in front of Jupiter or behind it, we can describe its projected side to side distance from the planet as zero (in any units). When the moon is at a right angle from the Earth/Jupiter line, it will be seen at its maximum separation from the planet. If you consider how it will appear from Earth as it orbits and moves between these two extremes, it will appear to trace out a sine curve from maximum separation, to zero, to maximum separation, to zero, and back again. Sine curves have a few basic properties:

  • They are repetitive. Their pattern repeats over time.
  • The maximum and minimum values of a sine curve are the same every time (i.e., the peak value is the same for every peak, which is also equal to the trough value). For the moons, this value is equal to its distance from Jupiter.
  • The distance between successive peaks is equal to the period of the sine wave. This is the length (of time, in this case) over which the pattern repeats. For the moons, this is equal to the amount of time it takes for them to orbit once around Jupiter.

Below is a sample plot for a fictitious moon of Jupiter. The x-axis is labeled Julian Date, which is an easier way of sequentially marking days than relying on our calendar, which is difficult to work with on a plot because the number of days per month varies. Each tick mark on the x-axis is one day. The y-axis shows projected separation from Jupiter in units of Jupiter diameters.

Plot of a sine curve to show how to find the orbital period needed to perform the calculation for Kepler's 3rd law.
Two-dimensional plot of the apparent separation between a fictitious moon and its host planet as a function of time
Credit: Penn State Astronomy & Astrophysics

If you again refer to the first image on this page, you will notice that when the moon goes from rightmost maximum separation to zero to leftmost maximum separation to zero to rightmost maximum separation again, that is the period of one orbit around Jupiter. On the curve above, the time from maximum to maximum peak is the same as the time from rightmost maximum separation to rightmost maximum separation. So, you can, therefore, estimate the period of a moon's orbit directly from one of these curves.

The amplitude of the curve illustrated above is the distance in Jupiter diameters when the moon is at either its left or rightmost maximum separation from the planet. That is a direct measurement of the semi-major axis of its orbit. Therefore, by plotting one of these curves, you can measure both P and a for a moon, which are the two quantities you need for a Kepler's third law calculation.

Recall that: P 2  = (4π   2  x a 3 ) / G(m 1  + m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So, therefore, if you have P and a measured, you get the sum of ( m 1  +  m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , which is in this case the sum of ( m Jupiter  +  m moon ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. .

On the next page are instructions for using a simulated observatory to take data on Jupiter's moons for the purpose of measuring the mass of Jupiter.

Simulation

We're going to use the animation below to simulate taking observations of Jupiter and its Moons. For this exercise, we will be observing one of the outer moons, Ganymede. You will see Jupiter located in the center of the screen, and four dots which roughly line up horizontally across the screen (this is the plane in which the moons orbit). You cannot always see all four moons, sometimes one or two will be directly in front of or behind the planet.

Directions

Start the Revolutions of the Moons of Jupiter simulator (below) by entering the following information into the appropriate boxes:

  1. Enter your birthday month and day into the Month and Day boxes. (This just allows us to randomize the data a bit, so each student gets slightly different measurements.)
  2. Enter last year into the Year box.
  3. Enter 12:00 into the Time (UT) box.
  4. Increments should be set to 24 hours.
  5. Click on the Begin button and the view will change to show you Jupiter and the Galilean moons.
  6. Each time you click on the Next Interval button, it will advance by one day and show you the appearance of the moons 24 hours later.
  7. Click on all of the visible moons, and the lower portion of the screen will report information like the date and time of the observation, as well as the name of the moon. One piece of data it reports is the Julian Day, which is a method for recording the date. If day 1 is Julian day 2454431, then the next day it will be Julian Day 2454432. It will also report the moon’s position (X and Y), as well as its distance from Jupiter in units of Jupiter diameters. If the number is negative, it means it is to the left of Jupiter and if it is positive it is to the right. Distance is the value you are most interested in and will be recording.
  8. For week 1 of this lab, please click on "next interval" 20 consecutive times, and record the date and the separation between Jupiter and Ganymede (record—and save—the information in the format of your choice: old fashioned paper & pencil, a spreadsheet program, or any other easily accessible format). If you get a "Cloudy Night," you can count that as one of your 20 data points, but be careful to account for this in your data table.

Animation caption:  Simulator that displays Jupiter and its Galilean moons as observed from Earth on a given date and a give time.


Source URL: https://www.e-education.psu.edu/astro801/content/l2.html

Links
[1] http://www.astronomynotes.com/
[2] http://www.astronomynotes.com/nakedeye/s15.htm
[3] http://solar-center.stanford.edu/AO/sunrise.html
[4] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Mars_prograde_2008.snf
[5] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Mars_retrograde_2003.snf
[6] http://apod.gsfc.nasa.gov/apod/ap031216.html
[7] http://apod.gsfc.nasa.gov/apod/ap011220.html
[8] http://www.astronomynotes.com
[9] http://www.astronomynotes.com/scimethd/chindex.htm
[10] http://www.astronomynotes.com/scimethd/s2.htm
[11] http://www.astronomynotes.com/scimethd/s6.htm
[12] http://en.wikipedia.org/wiki/Occam%27s_razor
[13] http://www.windows.ucar.edu/tour/link=/the_universe/uts/eratosthenes_calc_earth_size.html
[14] http://www.astronomynotes.com/history/s3.htm
[15] http://www.csit.fsu.edu/%7Edduke/nmoon6.html
[16] http://www.csit.fsu.edu/%7Edduke/moon6.html
[17] http://www.csit.fsu.edu/%7Edduke/nmercury.html
[18] http://www.astronomynotes.com/history/s4.htm#A3.3
[19] http://www.astronomynotes.com/history/s7.htm#A5
[20] http://en.wikipedia.org/wiki/File:Kepler-first-law.svg
[21] http://mathworld.wolfram.com/Ellipse.html
[22] http://en.wikipedia.org/wiki/Ellipse
[23] http://en.wikipedia.org/wiki/File:Drawing_an_ellipse_via_two_tacks_a_loop_and_a_pen.jpg
[24] http://en.wikipedia.org/wiki/File:Elps-slr.svg
[25] http://www.windows.ucar.edu/tour/link=/physical_science/physics/mechanics/orbit/eccentricity_range_anim_big_gif_image.html&edu=high
[26] http://www.windows.ucar.edu/tour/link=/the_universe/uts/kepler1_gif_image.html&edu=high
[27] http://phys23p.sl.psu.edu/phys_anim/astro/kepler_2.avi
[28] http://www.astronomynotes.com/gravappl/s2.htm
[29] http://www.astronomynotes.com/gravappl/s3.htm
[30] http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/NewtMtn/home.html
[31] http://phet.colorado.edu/
[32] http://galileo.rice.edu/index.html
[33] http://astro.unl.edu/classaction/
[34] http://www.teachersdomain.org/resource/phy03.sci.ess.eiu.galMoon/
[35] http://www.teachersdomain.org/resource/phy03.sci.phys.mfw.zweightlessness/
[36] http://www.teachersdomain.org/resource/phy03.sci.phys.mfw.freefall/
[37] http://phet.colorado.edu
[38] http://essp.psu.edu/sites/default/files/tes_esspissue.pdf
[39] http://galileo.rice.edu/sci/observations/jupiter_satellites.html
[40] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Jupiter_moons_lab_view1.snf
[41] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Jupiter_moons_lab_view2.snf