Measuring the Universe

 

The Cosmic Distance Ladder

Say calm. Numerical perspective is important for this FQ class. We are going to go a little deeper into some numbers. Please remember that it is the perspective and the inductive reasoning used to achieve that perspective that matter most.

First, watch this short video:

Measuring the Universe

Next read this BBC article:

It took centuries, but we now know the size of the Universe

 

Although there are many modern techniques for measuring the astronomical distances to stars, galaxies, and galaxy clusters (click here if you want a more complete summary), we will focus on the three mentioned in the video: parallax, standard candles, and Doppler red shift. Notice that these techniques are called the Cosmic Distance Ladder because like a ladder astronomers had to go "up" so to speak the learning curve, learning better techniques as the distances increased and previous techniques failed. Plus, each new measuring tool allowed astronomers to go "up" the ladder of greater distances, until out now with reasonable inductive confidence to the first galaxies formed shortly after the creation of the universe.

As we have seen, parallax was known to Ptolemy and astronomers in the Middle Ages. Interestingly, as we will see below, its limitations (it only works for relatively close astronomical objects) actually reinforced the view with "evidence" that the Earth had to be the center of the universe. So once astronomers realized that the sun was actually the center of planetary motion and not the Earth -- implying that the stars had to be much further away than thought by Ptolemy and people until the 16th century -- deep thinking caps had to be put on for applying the technique of standard candles. Then when even this method was found to be limited, the Doppler red shift method shown in the video was worked out.

Parallax

First remember your finger-eyes exercise. Along with the video and diagrams below, get the basic idea of parallax? Notice that after Copernicus and people realized that the sun was the center of our solar system (and not the Earth), the base of the triangle for the parallax measurement can be the diameter of the Earth's entire orbit, about 186 million miles, or two AU's as shown in the first parallax figure below. This increases the potential measured distance enormously for the parallax technique. The base line (A-B or 1-2 below) is increased 25,000 times from using different positions on the Earth as Ptolemy used. If instead of you finger, we used a short ruler, we could still use the parallax method to know when the ruler was closer or further by alternating the opening and closing one-eye and then the other. But if someone took the ruler and placed it a football field away, the ruler would not move against the background. We would not be able to tell if the ruler was 90 yards or 100 yards away. Given the distance, our base line (space between our eyes) is too small. But if we could increase our base line considerably, perhaps one person viewing the ruler twenty yards away to the right or left perpendicular to the line of sight to the ruler, then we could measure the distance of the ruler at different yard lengths away. So, now if a person observes a star at point A (or 1 in the second parallax diagram below) on the Earth, the same person can observe the same star at B (or 2 in the second diagram) six months later. (Remember that in a sun-centered system a year is one full orbit of the Earth around the sun, so 6 months for the Earth to move from point A to B (or 1 to 2 in the second diagram). As the author of the article linked above notes,

"Imagine having two eyes floating in space, either side of the Sun. Thanks to the Earth's orbit, that is exactly what we do have, and we can view stars' shift relative to objects in the background by this method."

Fascinating, and to their immense credit, pre-Copernican and even famous contemporaries of Copernicus, were aware that IF the Earth revolved around the sun, six-month comparative observations of stars could show parallax, providing dramatic evidence that the Earth was moving around the sun and not the sun around the Earth. Key point = The Earth-centered system predicts no parallax (because the Earth is not moving); the sun-centered system predicts parallax for the stars. The famous contemporary of and collaborator with Kepler, Tycho Brahe, was the greatest observational astronomer of his time. In the 16th century, before the telescope was used by Galileo, he manufactured and placed on the top of his castle, funded by his King, massive observational devices used to make precise measurements of the locations of the moon, sun, planets, stars, comets, and even super novas. Particularly important to the astronomers of his time was the accuracy of degrees and angular relationships of all astronomical bodies. For instance, it was important to know exactly where Mars was at any given time and date in comparison to say where the other planets were. As we have seen with Ptolemy's estimate of the moon's distance, a change in parallax angle would have dramatic results for a distance estimate. Consider this view of what is called a conjunction of the moon, Venus, Mars, and Jupiter:

Each astronomical object would have a pair of coordinates for the exact time and date this arrangement was viewed from Earth -- imagine vertical and horizontal lines crisscrossing this image, with some intersecting at the location for the moon, Venus, Jupiter, and Mars. It was important to have the precise coordinates for every astronomical body for every night to compare with the predictions of the different models of the universe. One of the motivations for having very precise data for the locations of the planets was that by the 16th century too often the predictions of both the Earth-centered model of Ptolemy and sun-centered system of Copernicus were off by a degree or more. (Imagine that both models predicted that Mars should be a little further away from the moon on this date and time.) In famous cases, predictions of some conjunctions similar to above were off by a month! Both Tycho and Kepler knew that a better astronomical model was needed. Even the Catholic Church was acutely interested in this issue and funded efforts in astronomy because an inaccurate astronomical model led to an inaccurate calendar, and it was of great concern that Easter appeared to be falling on the wrong date! If you believe that the Pope was the closest human being to God, you had better get Easter right. Tycho became convinced for reasons discussed below that a better Earth-centered model would be successful; Kepler was convinced that Copernicus was right, and a better sun-centered system would be successful. There was an obvious tension in their working relationship.

Interesting side story that was part of this tension: Kepler did not enjoy working with Tycho. Kepler was deeply religious; Tycho was an alcoholic and had wild aristocratic parties often. Tycho was rich and Kepler poor, in need of a job. Kepler also knew he had to work with Tycho, because Tycho had the best data in the world on planetary positions and motions. As with Copernicus and Galileo, Kepler was convinced that God had put the sun in the center of the universe and not the Earth. He needed though a sun-centered system that worked better than any Earth-centered system and for that he needed Tycho's data. Long story short: Tycho died suddenly -- he drank too much one night and his bladder burst(!) -- and after a big legal fight with Tycho's relatives, Kepler was able to get Tycho's data, eventually realizing that the planets move in elliptical orbits and not circles. Kepler essentially won the scientific battle, producing the solar system model we take for granted today, and showing that a sun-centered model could be much more accurate than any previous Earth-centered model. Important: he showed that even Copernicus was wrong -- the planets do not move in circles around the sun. Click here if you want to see the flawed (and very complicated) way Copernicus modeled planetary motion. Kepler thought that Copernicus had the sun's central position right, but he was also convinced the God would create a much more elegant universe than modeled by Copernicus. (Yes, religious views and one's view of God as the Master Mathematician played a major roll in the so-called Copernican Revolution. If interested in this part of the long story, see Chapter 5 of Science and the Human Prospect.)

For our purposes, here is the important point in this historical story. With his mighty instruments, Tycho was able to observe many stars at six-month intervals. He observed no parallax! Geometrically then there are two possibilities -- either the stars are very, very far away OR the Earth is the center of the universe. He calculated that if the Earth was really moving around the sun, the stars would have to be at least 700 times further away than the orbit of Saturn. His reaction? He could not believe that God would "waste" this much space, so the Earth-centered system must be true. He did agree with Copernicus though that the sun-centered system model produced some very elegant relationships, so Tycho supported what is called a geoheliocentric model of the universe -- the planets revolve around the sun, but the sun and moon revolve around the Earth! The Earth is still the center of the universe. If interested, do a Google search on "geoheliocentric model images."

So, in all the illustrations of parallax one finds on the Internet and in books, the important perspective to realize is that these illustrations are not to scale. The diagrams show parallax dramatically out of scale just to get the concept across. For instance, here are a few illustrations from the Internet.

 

The first image is from Cosmos -- The SAO Encyclopedia of Astronomy, and the second is from Chapter 5 in SHP. To scale, for the nearest star to our solar system, the star locations in these pictures would have to be over 15,000 miles away from the base lines (A-B or 1-2)!! Stay calm. As an FQ course it is important to understand the numbers below to some extent, but as a philosophy course it is more important to use your imagination. Imagine how small the parallax angle would be if we moved the stars in these pictures 15,000 miles to the right. Key point = for Tycho to discover that the sun-centered system was confirmed by measuring parallax for any star, he would have needed the capability of measuring an incredibly small parallax angle. Not possible without a modern telescope. A few decades later, even Galileo was not able to find parallax for any star with the new telescopes he had developed. The first parallax angles measured for any star (using 19th century instrumentation) was not accomplished for over 200 years after Tycho's efforts. With massive instruments, first on top of a castle and then later below ground to eliminate the slight movement of the castle walls, on just about every clear night for 20 years, he looked for parallax for about 1,000 stars. He found none and concluded that this was substantial evidence that the Earth did not move and must be the center of the universe. By today's standards, Galileo correctly argued that Tycho's and his own failure to find parallax implied that the stars were an incredible distance away. He did not have direct measurable evidence this was true? Plus, people had believed for over a thousand years that the universe was much smaller. Was Galileo endorsing the Copernican sun-centered system more on the basis of faith than evidence? The Catholic Pope at the time believed that he was, and finally banned Galileo from teaching and writing about the sun-centered system. If you are interested in this complicated historical story, see Chapter 5 in SHP.

Tycho was the acknowledged world leader in astronomical observational accuracy -- one of the reasons he was generously supported financially by the King of Denmark. His observations were 5 times more accurate than any of those by his competing fellow astronomers. How accurate? While most astronomers had been happy during the Middle Ages and early Renaissance period with a measurement of say a planet within one degree of its true position, Tycho was able to accomplish 1/30th of a degree routinely. But he would have needed to be able to measure an angle of parallax 1400 times smaller for the six-month movement of just the closest star to Earth (Proxima Centauri).

Let's stop here to do a little math and positional astronomy. Astronomers look at the night sky as a big round bowl -- a celestial sphere (not to be confused with the ancient notion of an ethereal heavenly substance) that can be divided (as with any circle) into degrees. In what is called positional or spherical astronomy, a celestial measurement is a location in the night sky expressed as declination and right ascension, analogous concepts to latitude and longitude coordinates on the Earth's surface. To note a location on Earth, we express the location in terms of latitude and longitude. For instance, Honolulu is located at 21.3069° N, 157.8583° W (° = degrees) -- the north and west mean north of the equator for latitude and west for longitude of what is called the prime meridian, a line from the north pole through Greenwich, England to the south pole. For the celestial sphere above us at night, a star chart might indicate that the bright star Sirius is located at Right Ascension 06:45:09 (6 hours, 45 minutes, and six seconds), Declination -16:42:58 (-16 degrees, 42 arcminutes, 58 arcseconds). In the picture above, all the astronomical bodies would have a location in terms of ascension and declination. For our purposes, again, just use your imagination. Imagine the lines of latitude and longitude ballooning outward from the Earth and being printed on the inside of the sphere of the night sky above. Viewed from Earth all the intersecting lines of latitude and longitude create a coordinate system in the sky. In positional astronomy all the intersecting points are described in terms of coordinates of declination and right ascension. Key point = accuracy of measurement was very important (especially for declination) for navigation (being lost at sea is not good) and also for providing the factual information to test different models of astronomy. At Tycho's time the big, big issue was "Is the Earth the center for the universe or is the sun?" A huge issue for Tycho and his contemporaries was "In a six-month period can any movement be detected in any star?" If a star actually moved in relation to the background but only a fraction of a fraction of a degree, and this movement was not detectable with the naked-eye instruments of the time, then what one was "seeing" gave the wrong answer to the question.

Here is what Tycho was looking for: A star's apparent relative movement in a six-month period.

 

Yet, for a thousand or so stars, all the right frames would be like the left frame. As careful, patient, and accurate as Tycho could be, no movement detected after 6 months.

So, let's see why numerically.

1 degree = 1/360th of a circle and can be divided into 60 arcminutes.

When the angles get very small, we have to use arcminutes and arcseconds:

1 degree = 60 arcminutes. Think of a round analogue clock being divided up into 60 minutes.
1 arc minute = 1/60th of a degree and 60 arcseconds or 1/21,600 of a circle (1/360 x 1/60). Think of a minute being divided up into 60 seconds.
1 arc second = 1/3600 of a degree (1/30 x 1/30) or 1/1,1296,000 of a circle (1/21,600 x 1/60)

Again, use your imagination --> one arcsecond = the angular size of a dime about 2 1/2 miles (about 4 kilometers) away!

Key number = number of arcseconds in a degree = 3600 (60 x 60 = 3600)

So Tycho was able to measure an angle of "only" 1080 arcseconds. (3600 x .30 = 1080). Punchline: Today we know that the parallax angle for even the closest star is .77 arcseconds. Notice the decimal point -- not a typo -- (point)77 arcseconds.

Given the level of accuracy he was capable of -- fantastic for a time before the use of a telescope -- he would only have been able to detect parallax for a star that was about two thousand Astronomical Units (1 AU = 93 million miles). 93 million x 2,000 = 186 billion miles.

Remember that the nearest star to Earth is about 25 trillion miles!

Due to the enormous distances we are now aware of, astronomers not only use the concept of a light year, but also what is called a parsec.

Here is an important relationship: 1 arcsecond of parallax = 1 parsec (3.26 light-years) In other words, if one can measure a star movement in six months of 2 arcseconds, the parallax angle is 1 arcsecond and the distance of that star is determined to be 3.26 light-years away. (See the two diagrams above. Notice the 2p in first one. We divide the angle in half because the angle we are interested in to do the trigonometry is the angle that goes with the right triangle as shown in the second diagram.)

Hence, just remember: 1 parsec = 3.26 light years.

So, today: If the parallax angle, p, is measured in arcseconds (arcsec), then the distance to the star, d in parsecs (pc) is given by:

d = 1/p

So for Proxima Centuari = 1/.77 = 1.30 parsecs = 1.30 x 3.26 light years = 4.2 light years = about 25 trillion miles away (4.2 x 6 trillion).

Remember that Tycho was capable of only 1/30th of a degree (1080 arcseconds). Even if he was capable of detecting 1 arcsecond of movement, he would have missed seeing the movement of the nearest star from Earth that only shows a parallax of .77 arcseconds.

Worth noting is that the brightest star in the night sky, Sirius, shows a parallax of 0.375 arcseconds, which gives us 2.7 parsecs using the simple formula. Hence, 8.8 light years (2.7 x 3.26) or about 54 trillion miles (8.8 x 6 trillion). "Intrinsically" (see the Standard Candle section below) it is not the brightest star in our galaxy. Although it is 26 times more luminous than our sun, it appears as bright as it does because it is relatively close to our solar system vantage point. See the picture below. Rigel is also very bright intrinsically (47,000 times more luminous than our sun), and the fifth brightest star in the night sky, but it is about 800 light years away.

Given the "facts" at Tycho's time, he felt he had very good reasons to believe the Earth did not move. One, we do not feel it moving. If it did move it would have to rotate on its axis at over a 1,000 miles per hour at the equator (it does,1040 miles per hour!). For people of Tycho's time, without Newtonian concepts of inertia and gravity, what would keep objects from flying off the Earth? Wouldn't this incredibly speedy merry-go-round movement cause a horrendous wind? Two, humans were special. Wouldn't God put humans in the center of the Universe? Doesn't the Bible tell us the Earth does not move? If the sun-centered system was true, then the stars would have to be incredibly far away. Why would God waste all this space!

Keep in mind that based on what we have studied about inductive reasoning, Tycho believed he had substantial induction by enumeration and corroborating higher-order inductions (background knowledge) for a belief that the Earth did not move. After 20 years of carefully looking for parallax for numerous stars, he surely did not have flaky evidence. But he was still wrong!

Speaking of the star Sirius, Tycho had one more fascinating "reason" for rejecting the Earth's movement. Some stars appear very bright in the night sky. If these stars were billions and billions of miles away, Tycho reasoned, to appear as bright as they do being this far away, the stars would have to be as large as the entire orbit of Saturn. Impossible Tycho reasoned. Our sun would have to be puny in comparison. Well guess what, many stars are indeed as large and in some cases even larger than our entire solar system! Check this picture out for perspective:

 

See it? One single pixel. And remember that a million Earths could fit within our sun. Notice how small Sirius appears in comparison to the Betelgeuse and Antares. And Sirius is twice as massive as the sun. When I first took an astronomy class many years ago, I did not fully understand why the instructor kept referring to our sun as a "yellow dwarf star." Pretty clear now. Betelgeuse and Antares are referred to as "Red Giant" stars. Giant indeed. Today our sun is classified as a G2 dwarf star.

 

See why? 1/.5 = 2 parsecs. 2 parsecs x 3.26 light years = 6.52 light years. 6.52 light years x 6 trillion miles = 39.12 trillion miles.

Standard Candles

Notice that Tycho appears to have made a back of an envelope estimate of how far a star would have to be from Earth based on its apparent brightness. Notice also that the method of trigonometric parallax is not going to get us very far measurement-wise out into the vast distances of a universe where even trillions of miles are a speck of space. From even the best ground based telescopes, the parallax limit is about .01 arcseconds, which allows us to detect stellar parallax for any star within 100 parsecs, 326 light-years, and 1,956 trillion miles. Remember that we are in a huge galaxy, and there are only about 1,000 "neighbor" stars within this distance. Using a satellite called Hipparchus, astronomers were able to achieve an accuracy of .001 arcseconds -- 1000 parsecs and 100,000 relatively close by stars. Remember that there are at least 100 billion stars in just our galaxy alone, and we also want to know the distances to many of the galaxies in our vast universe. Some people are curious. They want to know, really know, where we live and the location of our home in the universe.

So, as the video linked above demonstrates, believe it or not, we can increase our distance estimates enormously by examining the light (apparent brightness or magnitude) of astronomical objects, provided we know how bright these objects really are (intrinsic brightness). Let's examine the inductive evidence progression. First, we learn how light behaves on Earth. We learn a simple mathematical relationship called the inverse square law for light intensity -- the intensity of light observed from a source decreases the square of the distance from the object. We use 1/r². So, if we double the distance from a light source (1/2²), the intensity will be 1/4 of the original value.

The r above is the radius distance, knowing that light is a wave of energy emanating in a sphere from the light source.

Study this illustration:

So, the light that reaches Earth from a distant object has spread out over a vast spherical distance. Since the radius of the sphere is the distance to Earth, we can use the surface area formula for a sphere = 4πr² to get:

l = L/4πr²

Where l is the received intensity of the light and L is the original luminosity. If we know L and can measure l accurately, we can solve for r, the distance to the light source, as follows:

r =

(The equation editor of this html program is not very good. Inside the square root symbol ( √ ), reads L divided by 4 x π x l.)

The key point is that if we know the light power of L, and then can measure the apparent power (l) when the light is received on Earth, astronomers can compute distances much further than with the standard parallax method. Believe it or not, ordinary light contains a lot of information, and believe it or not there are instruments astronomers use to measure the apparent brightness of a light source. Do a Google search on a "bolometer" and/or "bolometric magnitude" for the complications addressed.

But how does one know the intrinsic brightness of a very distant object when one can surely not go there and see the object close up? Physics and math. Notice that the video and the article linked above mention special stars called cepheid variables. Think of a star that pulsates regularly with a cycle of brightness with a frequency related to its luminosity. Discovered in 1908 by Henrietta Swan Leavitt, astronomers and physicists now believe they understand these stars very well. There are different types but one class of cepheids range in size from 4 to 20 times the size of our sun and are 100's of thousands of times brighter. Long story short, these stars can be spotted with powerful telescopes in galaxies beyond our Milky Way and then the distance to these galaxies computed. For instance, in 1924 the famous astronomer Edwin Hubble used observations of these stars in the Andromeda galaxy to persuasively argue that Andromeda was another galaxy about a million light years from Earth. His estimate was off by over a million light years, but at the time a million light years and even the existence of another galaxy were shocking.

For an actual calculation, using another formula that involves the technical astronomical concept of magnitude, here is one from The Australia National Telescope Facility:

Formula used: [d = distance measured in parsecs; m = apparent magnitude; M = intrinsic magnitude]

d = 10^{(m - M + 5)/5}

Substituting some numbers for apparent and intrinsic magnitude: m = 15.57 (a relatively dim object compared to the sun, -27); M = -3.6

d = 10^{(15.57 - (-3.6) + 5)/5}

d = 10^24.17/5

d = 10^4.834

d = 68,234 parsecs

This measurement is for what is called the Large Magellanic Cloud (LMC), a satellite galaxy of the Milky Way. 68,234 parsecs (or 68.2 Kiloparsecs) = about 223,000 light years away (68,234 x 3.27 = 223,126). It turns out that there are two satellite galaxies of our Milky Way -- the Large and Small Magellanic Clouds. So Andromeda, at 2.5 million light years, is usually referred to as the closest "major" galaxy to our Milky Way galaxy. Use your imagination. Imagine our galaxy, a little smaller than Andromeda, spinning around like a giant frisbee, 100,000 light years in diameter, and then two satellite galaxies a couple of 100 thousand light years away. On a clear night in a nice non-light-polluted location, we can actually see these satellite galaxies. Here is a picture, courtesy of the National Geographic Magazine. The angled concentrated band of stars and dust from the top middle of the picture to the bottom left corner is the plane of our Milky Way galaxy. The two smudges of light on the right side of the picture are the LMC and the Small Magellanic Cloud (SMC).

For the original article from National Geographic, click here.

Cepheids are commonly used for distances from 1kpc to 50 Megaparsecs (Mpc). Or, 1.917¹⁶ to 1.91735116²¹miles, or 3,262 to 163,078,189 light years.

Ok, not bad, out to about 163 million light years. That will allow astronomers to compute the distance to a lot of galaxies, and even some clusters of galaxies. Do a Google search for pictures of the Virgo Cluster -- about 2000 galaxies and 65 million light years away. But we are only capturing a fraction of the estimated 100 billion or more galaxies believed to exist in the visible universe. Remember that the closest galaxies in the Coma Cluster are about 350 million light years away, and there is a lot going on from 163 million to 13.8 billion light years away! As the article linked above noted, we can get further using Type 1A supernovas that go off in distant galaxies as standard candles, but to get to the edge of our visible universe we need to use the Doppler Red Shift method.

Doppler Red Shift

Above it was noted that light carries a lot of information. In astronomy there is a saying, a "spectrum is worth a thousand pictures." The light from any light source can be analyzed not only for brightness, but also for what the emitting object is made of, how hot it is, and most important now, as the video noted, how the object is moving.

But wait. In referring to light, we are really referring to electromagnetic energy. Visible light is actually a small slice of the entire range of electromagnetic energy. Notice how small the visible portion (rainbow colors) is:

Notice we use a portion of the electromagnetic spectrum for our modern cell phones. All electromagnetic radiation consists of particles (we think!) but also it is a wave phenomenon. Notice the different wavelengths (distance between two wave crests). The fact that all electromagnetic radiation in general and light in particular manifests itself as both particles and waves is one of the biggest mysteries of our time. We will not have time to study what is called Quantum Mechanics and what is called wave-particle duality. Briefly though, imagine being at Waimea Bay on a big set day and watching 30-foot wave after wave close out the entire bay. But instead of seeing any particular wave flood the entire beach, these waves are mysteriously special. Every time an entire spread out wave barely touches the beach, the entire energy of the wave and the wave itself collapse at just one point on the beach and creates a big explosion of the concentrated energy that a split second earlier was spread out across the entire bay! Believe it or not, physicists have equations that describe a similar process for light waves. Electrons behave the same way. If interested do a Google search on "wave-particle duality," "collapse of the wave packet," and/or see Chapter 8 in SHP. There are also a lot of YouTube videos on "the two-slit experiment." Click here for one that is entertaining and fairly clear.

Amazing is that all our modern electronic technology uses quantum mechanics as a foundation. We have learned to use it even though we do not understand what kind of a reality produces it! As I write this, there are also incredibly futuristic technologies being worked on, including quantum encryption, which will make the Internet more secure, and quantum computing. Supposedly a single quantum computer would have more computing power than all the computers in the world today put together. xxx

All electromagnetic radiation results from atoms emitting photons. So with the right tools, astronomers can not only receive these wavelengths from astronomical objects, but infer a lot about the objects (because they are made of the atoms) that emit the radiation. A spectrum is actually worth millions of pieces of information. For very distant astronomical objects, the radiation is like a fossil telling us what happened billions of years ago.

Our focus, however, is not on all the other information we can unpack from light and other electromagnetic emissions. As the video noted, we can make a reasonable inductive inference that super distant galaxies are moving away from the Earth and our galaxy at incredible speeds, and the further a galaxy is away, the faster it is moving away from us. Think first about how much light we are receiving. For objects billions of light years away, the energy of all the observations probably amounts to less than a snow flake hitting the ground. Recall the Hubble Deep Field Image:

Remember that the dark empty space the Hubble Telescope focused on was about the size of a ping pong ball a football field away. Now imagine that techniques and tools exist for focusing on just one of faintest of these galaxies and analyzing the light that left it over 10 billion years ago!

Bottom line: When the light spectrum is examined, we find key markers of the light source substantially red shifted. This information can be qualitatively examined to not only demonstrate that the galaxy is moving away from our galaxy at an incredible speed, but then the distance "reasonably" calculated as well.

Let's examine the amazing inductive reasoning trail that allows us to infer such a profound and grand conclusion from such little received energy.

First, astronomers and physicists tell us that our sun is full of mostly hydrogen gas. How can anyone know that? The sun is 93 million miles away and no one can stand on the sun or anywhere near it to examine what it is made of. Well we can shine light through a container of hydrogen gas in a lab on Earth and then examine the spectrum of light that emerges. Similar to shining light through a prism. Here is a nice illustration form Wikipedia:

Go to the original site to see animation.

Long story short, shinning light through a gas, allows us know what the gas is made of. Here is what we see with a spectroscope (picture from the Khan Academy) when light passes through hydrogen gas:

If we see this same result when we examine light from the sun, we can reasonably conclude that the sun is made up of mostly hydrogen gas. There are lots of complications that are beyond the scope of our course; let's focus on the basic idea discussed in the video. Notice the absorption and emission lines. If an object is moving towards us, all the lines will be shifted more to the blue part of the spectrum. If an objective is moving away from us, the lines will be shifted more to the red part of the spectrum. What astronomers have seen over many, many decades is that the further away a star or galaxy is, the greater the red shift. Here is a comparison of the hydrogen spectrum of our sun with a cluster of galaxies.

From https://commons.wikimedia.org/wiki/File:Redshift_horizontal.png

Knowing the distance is actually a three-step process.

Step 1: First the redshift of the galaxy light is determined. Redshift = z

z = (λ_v – λ₀)/ λ₀

The symbol (λ, lamda) refers to wavelength, usually measured in Angstroms (designated as Å).

λv (lamda v) = the observed wavelength that has shifted.

λ₀ (lamba 0 = is the rest wavelength observed in a lab on Earth.

Example:

Suppose we measure a particular absorption line in Hydrogen from a distant galaxy to be 5010Å.

But the laboratory rest wavelength is 4861Å. So we have:

(5010 - 4861)/4861 = 149/4861 = .03065

Step 2: Determine the velocity the galaxy is moving away from Earth.

v = c * z (* = times; c = the speed of light)

v = 300,000km/sec * .03065 = 9,195 km/second

Notice a few points.

Here we use kilometers rather than miles. Easier because the speed of light gives us a nice round number -- 300,000km/sec. Remember that if you can understand (imagine it) better in miles, then we get 9,195 x .62 = about 5,700 miles per second. Notice please per second, not miles per hour. Imagine a huge galaxy moving away from our Earth and our galaxy at 5,700 miles per second! Stay calm, but everything is moving at an incredible speed. The pop-up above notes that the Earth is moving also. Snap your fingers and our solar system, and hence the Earth as well, have moved 150 miles from the point of view of the center of our galaxy. Even at this speed, our sun and planetary system will take 250 million years to revolve around our galaxy once. Remember that the Earth is spinning around (rotating) at the equator at over 1,000 miles per hour. It is also revolving around the sun about 67,000 miles per hour (18 1/2 miles per second). About 1,000 times faster than the speed limit on many U.S. freeways.

Next, notice that we got a positive number for z. If astronomers get a negative number, that means the absorption and emission lines are blue shifted and the object is moving towards us. Light from the Andromeda galaxy is actually blue shifted. Due to gravity, Andromeda and our galaxy will collide (merge actually) in about 4 billion years. (For a YouTube simulation and discussion, click here.) Andromeda is moving towards the Milky Way at 250,000 miles per hour -- the same speed that it would take us to get to the moon in an hour. The merger will occur about the same time that our sun will expand and engulf the Earth! Some of the galaxies in the Virgo cluster are also blue shifted, not because they will eventually collide with the Earth, but because the cluster is rotating and some of the galaxies have a motion towards our frame of reference.

The key point is that the vast majority of galaxies are red shifted. Major implication = the universe is expanding like a balloon. Imagine a balloon with a lot of Sharpie Pen inked dots on it. As it is inflated, every dot will move away from the other dots. The galaxies are not really moving and they are not "moving away" just from Earth. It is the space that is expanding, so observers on any dot (galaxy) will observe all the other dots moving away, creating the illusion that the dot is the center of the universe. But there is no center, just a universal colossal expansion. What is causing the expansion? News flash: we are living in the midst of an explosion, a Big Bang that occurred about 14 billion years ago. A universe that has expanded now to be about 93 billion light years in diameter. Long story, but it appears we are living at a very lucky time in the history of our universe, a relatively mellow time as the explosion dissipates. We also live at a lucky time in our galaxy and in a "Goldilocks" location for life. As real estate people say, it is all about "location, location, location."

But let's get back to the final step.

Step 3: Determine the distance of the galaxy. Here is the simple (but mind blowing) equation:

v = H₀d

So:

d = v/H₀

H₀ = the Hubble constant -- 70km/s/Mpc (explanation below)

So, the distance to our hypothetical galaxy:

d = 9,195/70 = 131 Mpc

Remember that one megaparsec = 1,000,000 parsecs, and one parsec is 3.262 light years, so 3,262,000 light years.

131 x 3,262,000 = 427,322,131 light years.

So, our hypothetical galaxy would be about 427 million light years away.

Let's not do the miles and/or kilometers, but if you want to play, multiply 427 million by 6 trillion for the miles or 9.6 trillion for the kilometers!

What is important for us is numerical perspective. This galaxy would be 171 times further away than Andromeda (427/2.5 = 170.8), but only about 1/28th out to the edge of the visible universe! (427 million/14 billion; approximately 1/2 of one billion, so 14 x 2 = 28) Use your imagination! Think it until you feel it.

Try one more amazing video: Morn 1415-Laniakea

The Hubble Constant

The estimate for this number is a very big deal. First, what is it?

Technically today it is seen as the number that describes the rate of expansion of the universe. It is a ratio of the recessional velocity of a distant object and its distance. For every million parsecs (1 Mpc, 3.26 million light years) away from us, a galaxy would be moving away from us at a particular velocity due to the expansion of the universe. Remember the balloon. All the dots move away from a particular dot, but distant ones would move away faster. Our hypothetical galaxy that is 427 million light years away would be moving away ten times faster than a galaxy that is 42.7 million light years away.

Today's best estimate of the Hubble Constant = 70 kilometers or 43 miles per second per megaparsec distance.

So our 427 million light year galaxy would be speeding away from Earth at about 5,632 miles per second (3,492 kilometers) and one only 42.7 million light years away would be receding about 563 miles (349 kilometers) per second.

Math = (427/3.26) x 43 = 131 x 43 = 5,632 (or 5,632 x .62 = 3,492)

There is a fascinating history to the Hubble Constant that even involves Albert Einstein. Einstein's General Theory of Relativity actually predicts the expansion of the universe, but at the time (early 20th century) Einstein did not believe this expansion was possible. So he put in an ad hoc "constant" in his equations that essentially stopped the universe from expanding. After Hubble discovered red shifts and the evidence that the universe was expanding, Einstein realized that the creation of his fudge constant was (he said) the "greatest blunder" of his career. If interested, do a Google search on "cosmological constant, Hubble constant."

The Logical Process of Scientific Method and Justification

For a course in inductive reasoning, at the end of the first short video linked above, the narrator makes a crucial point. She says that the most fascinating realization is "how all these (measurement techniques) build on each other." These methods for determining distances "fit together." The standard candle and red shift methods can be used to corroborate the parallax method. The parallax (for relatively short distances) and standard candle methods can be used to corroborate the red shift method. Similar to checking a math derivation several ways to make sure you have the right answer, we have greater confidence against the risk of being wrong when we get the same answer via independent methods. Even though we can only "go out" to about 163 million light years with the parallax method, if we apply the standard candle or red shift methods to objects within this range, we get the same result. This fitting-together-and-pointing-to-the-same-conclusion experience is fascinating for scientists, because getting the truth is so difficult in an uncertain world. When lots of different perspectives, assumptions, and experiences all point in the same direction -- when independent rigorous test after test gives the same result -- confidence is gained that nature is trying to tell us the objective truth. As the author of the above linked article also notes, ". . . the more ways of measuring distances we have, the better we can understand the true scale of our cosmic backyard."

Remember that countless randomized controlled studies show that smoking is harmful to lungs. Independently, scientists find that there are hundreds of carcinogenic chemicals in tobacco smoke. Even more confidence is attributed to biological evolution because numerous well supported scientific conclusions from many disciplines (astronomy, chemistry, genetics, geology, paleontology) fit together to support the general concept of biological evolution. Evolution is the foundational principle for today's biomedicine.

Similarly, in using any one of the deductive (mathematical) methods for astronomical distance measurement, we are making assumptions for premises along the way. If any of our premises are wrong, our deductive conclusion can be wrong. These methods involve independent assumptions. That is a huge plus from an inductive point of view. If we get the same general conclusions from three independent methods, we have much higher confidence that we are not fooling ourselves and nature is telling us something that is probably true.

Remember our murder-by-acquaintance example. Remember the concept of higher-order induction corroboration. If you were on a jury and heard testimony about the glasses of wine, dog not barking, no forced entry, and then also heard follow-up testimony that a murder suspect was not only in business with the murdered person, but that the murder suspect and the murdered person, according to testimony, had been fighting over a lot of money, that the finger prints of the murder suspect were found on one of the glasses of wine, and that the murder suspect was seen buying a gun the day before the murder, that the bullets that killed the person were traced to that gun, and the gun was found with the murder suspect's finger prints on the gun -- plus the murder suspect had no alibi for the time of the murder -- what would you infer? What if the murder suspect was also seen running down the street after the murder and throwing the gun in a trash dumpster? If you were on a jury, you would surely have evidence "beyond a reasonable doubt" that the murder suspect was probably guilty. Could he or she be innocent? Yes! Could we be wrong about the distances that show how big and old the universe is? Yes.

Science is an ongoing, constant checking and re-checking process, because the final, crucial logical process is based on inductive reasoning.

To understand the logical process, remember our barrel of apples example:

First we use deduction: If all the apples in the barrel are rotten, then in selecting any sample form any location we will find a rotten apple.

Then we test our hypothesis by selecting apples from different locations. Even if all the different tests match the prediction -- some apples off the top, then representative sample, etc. -- we are now using induction and generalizing that from "some" apples from our sample testing are rotten, we believe that all the apples are rotten.

Below we will see more realistic detail, but first let's see how this simple example is analogous to the logical structure of the overall method of science, often called the hypothetical-deductive method. From a hypothesis we deduce what should happen (predictions).

H --> E (evidence if our predictions are true) -- Deductive process: If H is true, then E should be true.

Then we use the successful predictions to generalize that our hypothesis is true.

E --> probably H -- Inductive process: If E is true, then H is probably true.

No matter how many Es we have, we still have only probability for H; we still have risk and can be wrong.

Simplified Hypothetical Deductive Method

 

Hypothesis

(deduction)

↓
↓

Prediction (Evidence if true)

 

Observed Predictions (Evidence)

(induction)

↓
↓

Hypothesis

 

 

Indeed, for every scientific conclusion there is the But What If We're Wrong possibility. This entertaining popular book by Chuck Klosterman uses a few examples that should ring a bell for us. Ptolemy and Tycho did excellent scientific work (mathematically and observationally), but they were wrong! Almost everything going on in their cultures reinforced that they were right about the Earth being the center of the universe. A hundred years from now will another culture look back at today's scientific beliefs and laugh? Could smoking cigarettes be found to be good for our health? [The comedian Woody Allen once made a movie (The Sleeper) about a health store operator waking up in the future to find that all his beliefs about health were wrong.]

But wait! Notice, inferring that because people were wrong in the past, therefore the beliefs of the present will also be wrong in the future, is also an inductive argument and one that attempts to predict the future! One that takes a risk and could be wrong. Perhaps the culture one hundred years from now will be building on what today's scientists believe to be true, just as the narrator in the film notes we have built up our view of the universe today. The truth will undoubtedly be bigger (more comprehensive), but what we believe today could still be a significant part of that truth. As we contemplate these beliefs and possibilities, there are astrophysicists that take seriously the possibility that our universe is just one universe in a multiverse -- an infinite sea of expanding bubbles with our universe being just one tiny bubble.

Let's be honest about how many things can go wrong and the real complexity of the deductive-inductive inference situation. Contemplate (stay calm) this graphic:

More Realistic Hypothetical-Deductive Logical Situation in Science

The Hypothetical-Deductive Method and the Web of Belief

This graphic shows the more accurate and hence more complex logical situation in science. Let's say that T1 is our current astronomical theory about how big the universe is. This theory actually involves many hypotheses (H1 . . . Hn), plus lots of assumptions or what are sometimes called auxiliary hypotheses by philosophers of science (A1 . . . . An). Then we use the sets of (H1 . . . Hn) and (A1 . . . An) as premises to deduce predictions about what we should see. If we see the predicted outcomes, these outcomes become evidence (E1 . . . En) for T1. (The double arrow, <-->, indicates that we first deduce predictions, -->, (potential E's, and then if the E's are confirmed we infer, <--, inductively that the sets, premises, for T1 are probably true.)

To name just one assumption, in the parallax diagrams above, we are assuming that it is permissible to treat the Earth's orbit as a circle, even though it is an ellipse. Notice the arrow to (2) in the realistic diagram. We believe that although the Earth's orbit is an ellipse and not a circle, it is a very "flat" ellipse and almost a circle. That conclusion is derived from its own set (2) of hypotheses, assumptions, and evidence. Every hypothesis and every assumption has its own trail of reasoning and inference justification, any one of which could be wrong. Get the big picture? When we test a theory, we don't just test one hypothesis or belief, we test an interconnected web of beliefs.

Remember please an important logical fact about deductive reasoning. If one has a set of premises (H's and A's) and we infer the valid conclusion (a prediction E), and the prediction is false, this only proves that at least one premise (one H or A) is false. Valid arguments with false conclusions have at least one false premise. Initially supporters of the sun-centered system were confused by the fact that no one could find any parallax for any star. The sun-centered system predicts parallax. For Tycho, along with other reasons, this was enough to refute the sun-centered model. But wait. We can "save" the sun-centered system logically and make it consistent with the evidence (no parallax) by adding an A -- the stars are very, very far away. Let's call this an auxiliary-save -- we are saving the logical problem of apparent refuting evidence by making the evidence consistent with the theory by adding a supplementary hypothesis.

Are auxiliary-saves just excuses, fudge factors, and logical tricks? Truthiness patches or truthful hyperbole? Recall the example of the person who believed he had pictorial evidence of ETs on Earth. But all the pictures were blurry. Enter the auxiliary-save (lame excuse?) that the ETs sent out special radiation whenever pictures were taken to make the pictures blurry. The sun-centered auxiliary save turned out to be true (200 years later!), but at the time the Earth-centered model supporters treated it as a lame excuse similar to the blurry picture save.

Some of the Earth-centered supporters also did their fair share of auxiliary-saving. When Galileo made some of the first telescopes, the instrument was so new that one could doubt that what it revealed was real. So Galileo would do demonstrations on viewing objects far away on Earth. He would show a group of skeptics an object A through a telescope, the details of which were not visible by the naked eye, and then they walked over to A and saw the details up close revealed by the telescope from afar. But wait said the skeptics. Just because the telescope works on Earth does not mean it works viewing the celestial spheres and the different levels of heaven! Galileo was able to show with the telescope new stars that had never been seen before, showing that new astronomical objects could be discovered in the celestial realm. (He also showed that Jupiter had moons that were not seen before and were not supposed to exist.) The traditional view was that the heavens were perfect and unchanging. God does not make any changes to the heavenly spheres. So, if the telescope was showing things that were not supposed to be there, they reasoned, then this is evidence that the telescope does not work when pointed "up" towards heaven!

We can now define relativism more precisely. According to the relativists, truthiness and truthful hyperbole by adjusting our webs of belief are what we all do. Hence, remember that the relativists reject one of the major assumptions of this course -- that we can reason about risk and demonstrate that some beliefs have more inductive support than others, that some beliefs are more probable, more likely to be true, and hence more reliable to follow. Essentially we can see now that the relativists believe that any web of belief can be saved and protected from evidence by simply tinkering and adjusting premises in the web. Don't some politicians make logical auxiliary saves all the time? "Oh, what I meant was . . . ". "You are forgetting that I also believe . . . " Shouldn't we be wary when someone has an answer that he or she is right no matter what happens? No matter what the evidence is?

The rejection of philosophical relativism is one of the most important messages in this class. Think about what we have just covered. Relativism is wrong, but we should not underestimate the sophistication of its appeal. Logically it correctly zeros in on the uncertainty of all the nodes in a web of belief and all the perspectives one can use in attempting to describe reality. However, ultimately it takes advantage of the many nodes of uncertainty in the full H-D logical situation and draws the wrong conclusion (all beliefs are equal in probability) from uncertainty and risk. For those who believe in objective truth, the potential weaknesses are also strengths and what the narrator was excited about at the end of the short video. When you keep testing all the nodes and you do so in a lot of different ways and the overall picture continues to fit together, one has more confidence that we are seeing not just how a web of belief fits together, but how objective reality also fits together. The truth is that many webs of belief will begin to fall apart when tested rigorously. One can always continue to try to patch with auxiliary-saves, but at some point, one begins to see that there are too many holes in the dike so to speak. Example: How could Noah realistically put two of every animal on his Ark? There are thousands of species of dinosaurs alone and some are as tall as six story buildings? Auxiliary-save = he put only babies on the Ark. Really? And how did he manage to retrieve the babies from the huge, dangerous, and protective parents? How did Noah manage the feeding of all these creatures? How did he manage the excrement!? For the latter, auxiliary-save = he built ramps (like narrow ski jumps) and trained the animals to defecate in the ramps so the excrement would flow off the boat into the water without any human maintenance. At some point one sees the saves becoming more and more implausible. The logic might work (one can have a lot of false premises in valid arguments), but the web becomes too flimsy (too many implausible node premises) for a rational person to accept. No matter how implausible the saves, they might be true. But mere possibility does not equal probable truth.

A long story that we cannot follow (again, if interested, see Chapter 5 in SHP); it is true that Galileo did not have the factual proof that the sun-centered system was true. He did not have persuasive evidence or high probability that the stars were an enormous distance away from our Earth and sun. (He had some evidence -- the observation of new stars, implying that if one could see further into the heavens one would see new astronomical objects and hence there was a potentially vast universe out there.) He also had a false theory to explain the tides and no theory of gravity. He also believed incorrectly (as did Copernicus) that the planets must move in circles. However, there was a "fitting together" of the sun-centered system that impressed even Tycho, and this fitting-together was enough to keep supporters working until Kepler produced the elliptical orbit break through.

But wait, one more item of honesty about uncertainty and risk.

Notice also the arrow from the E1 . . . E5 . . . En. News flash: There is no such thing as a "brute" fact. In science (and in life) we cannot just assume with certainty that everything we see is accurate (true). Do we really see that six-month movement of Proxima Centauri to be .77 arcseconds? That "observation" is based on lots of astronomers agreeing on the same assumptions and then seeing the same result. It is not based on just one person saying this is what he or she recorded. In a very real sense, even so-called facts are also inductive generalizations -- we think we see X based on some people confirming that they see X. Can there be controversy and disagreement about what people see? Yes. Remember the blue-dress Internet controversy? Is the dress blue and black or white and gold? Exactly why scientific observations are checked and rechecked, argued and reasoned about until consensus is obtained.

Take Away Message

Seeking objective truth is complicated and difficult. Philosophically science is based on empiricism -- "seeing is believing," what is true should be based on public observational experience. No result in science is accepted unless that result is based on lots of observational tests. We have to observe the results of randomized controlled studies. We have to see fossils and the results of DNA analyses. We have to see red shifts, different magnitudes for stars, and arcsecond angles for astronomical objects over a six-month period. Empiricism is a fallible method of achieving truth -- we can always be wrong, and we only have various levels of probability for inductive conclusions. But scientists do not just go to a bar, have a few drinks, and then make up all the numbers and observations we have covered. Although we could be wrong, the evidence is overwhelming that we really do live in a gigantic and very old (by human standards) universe, and it is not true that we live on a flat disk with a dome-like firmament holding back the waters of heaven, or that heaven is about 80 million miles away and just past the plane of a thousand stars all at the same distance from the Earth in the center of the entire universe. We have very strong inductive evidence and high probability for the reliable scientific conclusion that the Earth is not flat, not in the center of the universe, and that we live on a relatively small planet that revolves around a dwarf star, that revolves around an average galaxy containing at least 100 billion stars, and that this galaxy is just one of at least 100 billion galaxies in a very old and large universe.

The famous astronomer and science writer Carl Sagan said it best. We live on a (Horton-Hears-a-Who) small pale blue dot. Here is a famous picture of our Earth from 4 billion miles away taken by the Voyager 1 spacecraft in 1991 as it was leaving our solar system.

This is our home folks and remember beliefs matter. They matter for our values. What have we been doing while on this little oasis planet? What have we been thinking? According to Sagan,

"The Earth is a very small stage in a vast cosmic arena. Think of the rivers of blood spilled by all those generals and emperors, so that, in glory and triumph, they could become the momentary masters of a fraction of a dot. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner, how frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds. Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light." From Pale Blue Dot: A Vision of the Human Future in Space, 1994.

Quiz

 

 

 

 

 

Hard one. See above. Even one arcsecond would not have allowed Tycho to detect Proxima Centauri's six-month movement. One arcsecond would have only allowed him to see parallax for any star 3.26 light years away. Proxima is 4.2 light years away and shows a .77 arcsecond of movement.

 

 

 

 

 

Hard one? Let's do the math. Remember the formula = d =1/p. So for Rigel we have d = 1/.0042 = 238 parsecs, or about 776 light years (238 x 3.26 = 775.88). For Betelgeuse, we have d = 1/.0051 = 196 parsecs, or about 640 light years (196 x 3.26 = 638.96). Remember, use your imagination and some numeracy. The smaller the parallax angle, the further away the astronomical object. .004 is smaller than .005. Interesting in the light of the ongoing nature of science, if you search online for the distance estimates of these stars, you will find an ongoing discussion and revision of the estimates, within 20-40 light years. But no one believes Betelgeuse is closer than Rigel.

 

Remember, the brighter an object, the lower the number. The sun is -27.

 

 

 

 

 

 

 

Summary

See the Math Summary, next in the table of contents.

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