It is time to put together all the knowledge from the previous topics, and understand how those celestial objects that you can observe behave and why they behave the way that they do. it is called Astrophysics.
We will not, of course, go deep into this field, but there are a few itens you must at least understand.
Naked Eye Observations:
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.
The position of the Sun in the sky appears to drift with respect to the background stars (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.
If You want to check it out by yourself, there is more information on helical risings from Stanford on their site: Ancient Observatories, Timeless Knowledge.
As you can see from the list above, 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.
Below is a composite image of Mars' path "retrograde loop" between 2007-2008.
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!
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. In many cases the model is simply an idea—that is, there is no physical representation of it. However, that doesn't preclude us from making a physical representation of the model. 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.
This circular path of
- making observations,
- creating a model (or hypothesis) to explain these observations,
- making predictions for what future observations will show,
- making new observations to test these predictions,
- revising the model to explain the new observations, and
- making revised predictions,
can be repeated over and over again until you converge on a well tested explanation for your object of study. This process is usually referred to as the scientific method, and is the basis for all of the concepts presented in this course.
The last point I should mention, though, is that we are often taught that this is a linear, step by step process as I laid it out in the paragraph above. What I hope will be made clear in the rest of the course, though, is that in practice this process is very non-linear.
The Geocentric model:
Traditionally in Astronomy text refers to the ancient Greeks. Following that tradition we will study 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!
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.
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. As we have seen before, the retrograde motions of the planets are very complex; therefore, Ptolemy had to create an equally complex model in order to reproduce these motions. 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.
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:
- Earth (unmoving; located at the center)
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. 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:
The geocentric model of the Solar System remained dominant for centuries. But, because even in its most complex form it still produced errors in its predictions of the positions of the planets in the sky, scientists acknowledged that a better model was needed.
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 a Penn State University animation.
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.
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, using instruments he designed and built, Brahe compiled a continuous list of accurate positions on the sky for the planets over approximately 15 years. 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 of Planetary Motion:
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! Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits, Newton will develop a dynamic description that involves the underlying influence (gravity)
- 1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus (the other focus is empty).
Ellipses that are highly flattened have high eccentricity. Ellipses that are close to a circle have low eccentricity.
In the image above, the larger the distance between the foci (green dots), 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.
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.
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!
- 2nd law (law of equal areas): a line connectiing the Sun and a Planet (called the radius vector) sweeps out equal areas in equal times.
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.
Click on this image to launch the animation.
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:
- Objects travel fastest at the low point of their orbit (perihelion), and travel slowest at the high point of their orbit (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.
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 planets "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.
- P2 ∝ a3
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, too. The proportionality sign in the above equation means that:
- P2 = ka3
Where k is a constant number. If we divide both sides of the equation by a3, we see that:
- P2 / a3 = k
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:
- (P2 / a3) Earth = (P2 / a3) Mars = (P2 / a3) Jupiter
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 aEarth = 1 AU. If you plug 1 year and 1 AU into the equation above, you see that:
- (P2 / a3) Earth = (1 year)2 / (1 AU)3 = k = 1
So for every planet, P2 / a3 = 1 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:
- (P2 / a3) Saturn = (29 years)2 / (a AU)3 = 1
- (a AU)3 = 841
- (a AU) = 3√ 841 = 9.4 AU
So Saturn is 9.4 times further from the Sun than the Earth is from the Sun!
The 3rd law is used to develop a "yardstick" for the Solar System, expressing the distance to all the planets relative to Earth's orbit by just knowing their period (timing how long it takes for them to go around the Sun).
click here to see the inner SS orbits
click here to see the outer SS orbits
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:
- 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.
- 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).
- For every action, there is an equal and opposite reaction.
and also his law of universal gravitation:
- The force of gravity between two masses is: F = G (m1 x m2) / d2
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! The image below links to an animation that demonstrates Newton's Cannon:
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 a Penn State University animation.
Here is a simulation of a galaxy using Newton’s law of universal gravitation.
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 P2 = ka3. 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:
P2 = (4π 2 x a3) / G(m1 + m2)
Which means that k = 4π 2 / G(m1 + m2)
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, m1 = mEarth, and m2= mSun, so just by measuring PEarth and aEarth, you can calculate mSun + mEarth!
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, that is,
F = G (m1 x m2) / d2
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, 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:
- Write out this equation one time for the situation on Earth, that is:
FonEarth = G (m1 x m2) / donEarth2
- Write out this equation a second time for the situation in Space, that is:
FinSpace = G (m1 x m2) / dinSpace2
- Form a ratio taking the equation from #1 above and putting it over #2 above, that is:
(FonEarth) / (FinSpace) = (G (m1 x m2) / donEarth2) / (G (m1 x m2) / dinSpace2)
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:
(FonEarth) / (FinSpace) = (1 / donEarth2) / (1 / dinSpace2)
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. Fill in these values for donEarth and dinSpace 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.
The mathematical formulation of Newton's dynamic model of the solar system became the science of celestial mechanics, the greatest of the deterministic sciences.
Although Newtonian mechanics was the grand achievement of the 1700's, it was by no means the final answer. For example, the equations of orbits could be solved for two bodies, but could not be solved for three or more bodies. The three body problem puzzled astronomers for years until it was learned that some mathematical problems suffer from deterministic chaos, where dynamical systems have apparently random or unpredictable behavior (see below).
Differential Gravitational Forces (Tides):
Tides are caused by the interaction of a body's motion around a planet or the Sun and the internal force of gravity.
Water tides are caused by the fact that the water on the Earth's surface is more easily deformed by tidal forces than the rocky crust. And the strength of tides are dependent on three factors:
- location on the Earth's surface
- orientation of the Sun and the Moon (both has approximately equal tidal influence on the surface of Earth)
- geographic features (shape of bay, inlets, etc.)
What happens when the tidal forces become greater than the internal gravity of an object? The object is torn apart. This occurs when a moon approaches too close to its primary, a point called the Roche limit. The tidal forces increase as R, the distance between the planet and the moon, becomes smaller until the moon is disrupted into numerous small bodies. This is the origin to the rings around Saturn and other Jovian worlds.
Three Body Problem and Complexity:
Deterministic laws, such as Newton's laws of motion, imply predictability only in the idealized limit of infinite precision. The Universe itself cannot know its own workings with absolute precision, and there cannot predict what will happen next in every detail. Deterministic chaos seems random because we are necessarily ignorant of the ultrafine details and so is the Universe itself.
The behavior of complex systems is not truly random, it is just that the final state is so sensitive to the initial conditions that it is impossible to predict the future behavior without infinite knowledge of all the motions and energy (i.e. a butterfly in South America influences storms in the North Atlantic).
Even games with simple rules can produce complex behavior, as shown in the following example:
Although this is 'just' a mathematical game, there are many examples of the same shape and complex behavior occurring in Nature.