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

What does quarter moon mean?

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).

Right triangles and distance ratiosThis 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/DsunProblem– 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!

The relative distances of the Sun and Moon

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 89.85 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!

Angular sizes—relative distances and relative sizes.

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 theSun and Moon are almost the same!

Total eclipse, July 2010

Angular sizes versus angular distances

But, if the angular sizes are the same, then the ratio of the distances is the same as the ratio of the sizes!

Sizemoon/dmoon = A = SizeSun/dSun

Sizemoon/SizeSun=dmoon /dSun

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)

The size of the Sun, and the heliocentric Universe.

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…

The distance to the Sun—the astronomical unit.

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.

Switching on the light—High school Activity 2: Overview

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.

Starting at Earth. 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

Activity 2, subactivity 1– The inverse square law.

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

Inverse Square LawThe 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/(4p d2)

How light spreadsLight 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=4pr2

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.

Standard candlesOf 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.

HS Activity 2, sub-activity 1: Using light to determine distances

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

Activity 2, sub-activity 2—putting it into practice.

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—what are they?

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

Why do Cepheids pulsate:more information than you need

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….

Cepheid variables—why do they matter?

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 period-luminosity relation (1)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 2.512 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 1st century BC. Hipparchus called the brightest stars in the sky 1st magnitude stars, the next brightest 2nd magnitude, and so on until the faintest were 6th magnitude.

In modern magnitudes, the Sun is -26.7, and the faintest galaxies ever detected are at +30 or so

The period-luminosity relation (2)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 = -2.81 log(P) – 1.43

The Luminosity can be measured from the period!

Hubble and the distances to galaxies.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

HST and the distances to galaxies. 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 100th 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

The cepheid light curves.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 fromm – M = 5 log(D) - 5

Possible extensions. 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.