Measuring what you can't touch - from the size of the Earth to the structure of the solar system, black holes and the expansion of the Universe—session 2 Middle and High School investigations in astronomy

Quarter Moon is a special phase. At quarter Moon, exactly ½ of the sunlit half of the Moon is visible. Aristarchus new what this meant– it meant that if you thought about the triangle connecting the Earth, Sun, and Moon, the angle at the Moon is a right angle (90 degrees).

This is the same type of triangle as the ones we studied in the first activity! Looking at the figure below, Sin(A) ~ A/57.3 = Dmoon/Dsun Problem– the angle A is the angle at the Sun, not at the Earth! But, we know something else about triangles. The sum of the angles of a triangle is 180 degrees. If the angle at the Moon is 90 degrees, the angle at the Sun is 90-B degrees when B is the angle at the Earth!

So, the ratio of distances is Dmoon/Dsun = (90-B)/57.3 Aristarchus measured the angle B—the angle between the Moon and The Sun at quarter Moon—he found it to be 87 degrees. He calculated that the Sun was 19 times farther than the Moon. It’s a hard to measure the angle, because it’s hard to tell when the Moon is exactly at quarter. The true angle is close to degrees—the Sun is about 380 times the distance of the Moon– by making an error of less than 3 degrees, Aristarchus was wrong by a factor of 20!

What about the relative sizes? Well, Aristarchus also knew that there are total solar eclipses—and that they only last only a short time. The angular sizes of the Sun and Moon are almost the same! Total eclipse, July 2010

But, if the angular sizes are the same, then the ratio of the distances is the same as the ratio of the sizes! Size moon /d moon = A = Size Sun /d Sun Size moon /Size Sun =d moon /d Sun Aristarchus calculated that the Sun was 19 times bigger than the Moon—given the size of the Moon, this meant that according to him the Sun was ~7x bigger than the Earth. (Actually more than 100x)

Aristarchus, having shown that the Sun was bigger than the Earth, pointed out that it made no sense for the bigger thing to go around the smaller—the Earth had to go around the Sun. In 280 BC, he had the right idea. But he was not believed! Why? Partly, it was because there didn’t seem to be any parallax…

Using the technique of Aristarchus, we’ve measured the distance between the Earth and the Sun—this quantity (about 150 million kilometers) is a very useful basic unit for measuring distances in the solar system. It’s so useful it’s been given a name: the Astronomical Unit. We’ll see it again in activity 3.

The high-school astronomy GSEs center around ESS-3, which deals with the evolution of stars and galaxies and the Universe. Although parallax gets the distances to millions of stars, the angles for more distant objects are too small to measure. We need another way to measure distances. The second High School Activity introduces the second way astronomers measure distances—using the properties of light.

 We begin at Earth, by studying light from a campfire.  Study light intensity changes with distance—use sensors to uncover the inverse square law.  By determining the relationship “here on earth”, we can apply the concept to light traveling from stars

You are all familiar with the concept. Light sources that are farther away appear fainter.

The amount of light we detect falls off as the square of the distance. The amount of light a detector receives from a source per second over a fixed area is the flux The amount of light the source emits per second is the luminosity The relation between flux and luminosity is: F = L/(4  d 2 )

Light from a source that emits in all directions (isotropically) spreads out uniformly. At any given point in time, it covers a sphere whose radius increases with time (it’s increasing at the speed of light). That same light therefore is spread out over a larger area. The surface area of a sphere is Area=4  r 2 Therefore, the energy per area falls b y the same factor Each unit of area has less light passing through it when it is further from the source than when it is closer to the source.

Of course, the relation between flux (which you can measure) and distance doesn’t work if you don’t know the luminosity. That means you need to have Standard Candles In the first sub-activity, you will be working with a standard candle (standard lightbulb?)– In the second sub-activity, you will study a class of stars that can be used as standard candles.

 Our sensor activity  Uses a light sensor attached to the go- link  There are 3 settings on the sensor, so you’ll need to “play” to figure out the optimum settings based upon the light source you use

Now that we’ve learned about the inverse square law, we can put it to use to measure the distances to stars. First, we need to find some standard candles. In the first activity, we learned that stars aren’t all alike. So we need some special stars. We need Cepheid variables

Cepheid variables are stars that pulsate. They are constantly growing and shrinking, with very regular periods. As the stars pulsate, they get alternately more and less luminous The flux at Earth keeps changing

The stars pulsate because of opacity changes. In the interior of the star, there is a layer where the helium gas in the star is doubly ionized. The ionized gas absorbs more light, and this heats the gas. The hot gas expands, making the star bigger. But as gas expands, it cools. When the gas cools, it also becomes less ionized. This decreases the light absorption, and the gas cools so fast it contracts, contracting the star. The contracting gas heats, and re-ionizes, and the cycle starts all over again….

Cepheids aren’t all the same luminosity—some are brighter than others. But, they do have an interesting property, discovered by Henrietta Leavitt in 1908 during her study of the Magellanic clouds: There is a relation between the period of the Cepheids and their luminosity Photo Credit: AAVSO

The actual data from Leavitt’s 1912 paper The “modern” magnitude system assigns a magnitude of 0.0 to some flux (traditionally the flux of Vega), and then each magnitude difference corresponds to a factor of in flux (5 magnitudes is a factor of 100) What Leavitt found was that in the SMC, stars with longer period were brighter. The data is on the right. The x-axis plots the logarithm of the pulsation period, in days (so 0.0 is 1 day, and 2.0 is 100 days). The y-axis plots the apparent magnitude of the stars in the photographs. Magnitudes is a system of measuring the flux from astronomical objects astronomers have carried around with them since Hipparchus in the 1 st century BC. Hipparchus called the brightest stars in the sky 1 st magnitude stars, the next brightest 2 nd magnitude, and so on until the faintest were 6 th magnitude. In modern magnitudes, the Sun is -26.7, and the faintest galaxies ever detected are at +30 or so

The actual data from Leavitt’s 1912 paper The apparent magnitude (m) measures the flux– which depends on distance! There is an equivalent magnitude, called the absolute magnitude (M), which measures the luminosity— actually it measures the flux you would measure if the object is at a fixed distance. The diffference between m and M depends on the distance—if d is in parsecs: m – M = 5 log(d) – 5 The Sun has M~+5 in this system, while the most luminous quasars discovered have M~-24. The period-luminosity relation has been calibrated using the parallax measurements of the Hipparcos satellite: M = log(P) – 1.43 The Luminosity can be measured from the period!

The Cepheid variables were the key to measuring the distances to galaxies:  They are bright enough to be seen in nearby galaxies  Their luminosity can be determined from the period—they are standard candles. In 1924, Edwin Hubble used the then biggest telescope in the world (the Mt. Wilson 100” telescope) to measure cepheids in a nearby galaxy-- Andromeda

 The Hubble space telescope was built partly to measure Cepheid variables in nearby galaxies (it was one of its “key” projects). Putting a telescope in space makes finding individual stars much easier (less blurring). With the space telescope, we have been able to measure Cepheids in tens of galaxies. You will be using the data collected by HST on one of the galaxies, M100. (The 100 th object in Charles Messier’s album of fuzzy things that were not comets! M100 is a beautiful spiral galaxy. From the ground, individual stars are hard to spot. With Hubble Space telescope, it’s just possible. Credit: ESO Credit: HST

You will measure periods and magnitudes for 12 Cepheids measured by Freedman et al. (1994) The period will be used to infer M, then the distance will be calculated from m – M = 5 log(D) - 5

 Once you have the distances to galaxies, this can be the starting point for investigating the properties of galaxies (types, rotation, stellar populations)  Another possible place to go is to investigate how we measure distances to even farther objects (where single cepheids can’t be seen)—this leads to the topic of the distance ladder.