Measuring the Sky

Earth Science Extras

by Russ Colson

 

Hubble Space Telescope Deep Field View of Distant Galaxies

 

Recap of Lecture

 

 

Introduction

Back in the late 1700s and early 1800s, some science philosophers reasoned that we could never know, in a scientific sense, some aspects of stars, such as what they are made of. After all, science is based on observation, and the stars were much too far away to ever visit and study through observation. So, how could we ever know what stars are made of?

However, then came the invention of the spectroscope, and we gained a way to observe what stars are made of even from a great distance. The spectroscope splits light into its different wavelength components, and since each element emits its own unique wavelengths of light, we can figure out what stars are made of by studying the individual wavelengths of light coming from the stars.

Even without a spectroscope, it's possible to use the color of stars to learn about them. Just like a glowing flame has different colors at different temperatures, so too does star color indicate the temperature of the star: cooler stars are red, medium stars yellow, and hotter stars blue. If you go out in the country where there is less light pollution, you can see the differences in color. The constellation Orion, one of the most prominent constellations of the Northern Hemisphere winter sky, has a blue and red star--try to find them!

Our focus for this lesson is not on the composition of stars, or their temperature, but on how far away they are. How can we measure the distance to something when we can't run a tape measure out to it?

We are going to look further at two methods for measuring the distance to stars or galaxies, both introduced in the lecture. These are:

1) Parallax

and

2) the Standard Candle method

 

Parallax and Measuring the Distance to the Stars

Thumb Parallax-- Understanding the Lecture

 

You can think of the apparent shift in the position of your thumb as outlining a traingle, as shown below. We can split this triangle into two right triangles. The parallax angle is half the angle of the perspective shift, or the angle of one of the right triangles.

 

Of course, in reality, your thumb is not moving; it is your perspective that is moving. So the reality is better illustrated by thinking of a triangle marked by the position of one eye, the position of the other eye, and the position of your thumb, as shown below. We can again split this triangle to get two right triangles (as shown). One of the amazing and interesting aspects of right triangles is that, if we know the length of one side and the value of one angle, we can calculate all the other sides and angles! For example, if angle β in the illustration is 75 degrees, then angle α is 15 degrees (α = 90 - β). The lengths of sides and the angles are related to each other by the mathematical rellationships shown in the image. In this illustration, α is the parallax angle and H is the hypotenuse of the right traingle.

 

While the parallax shift of your thumb in the lecture is caused by the shift in perspective from your left eye to your right eye, the shift used to measure the distance to stars results from the shift in perspective from the Earth's location in one season (say, January) to its position 6 months later when it is on the other side of the Sun (say, July). We can observe the apparent shift in a star's position relative to more distant background stars as shown in the illustration below.

 

 Again, we can imagine a right triangle defined by the apparent shift in the position of the star (shown in red in the illustration above). If we know that the distance between the Earth and the sun is 1AU (meaning one astronomical unit, which is about 93 million miles) and if we measure the parallax angle, then we can calculate the distance to the star using geometry, or the trigonometric functions given above.

 

 

 Parallax Measurement of a Real Star

When we look at stars through a telescope, and take an image of stars on film or digital device, the position of the stars are translated onto a flat image. The parallax shift shows up as a shift in position on this image, as shown in the illustration below. More distant stars will shift less while closer stars shift more. The most distant stars will have no detectable shift in position.

 

 

The two images below were taken 6 months apart (thus the Earth has moved from one side of the Sun to the other side, a distance of about 2AU). The star HT Cas is the closest to Earth of the stars in this image. You can identify it in the image by figuring out which star has moved the most relative to the other stars. Note: This activity comes courtesy of the NASA Goddard spaceflight center, with teacher activities overseen by Dr. Barbara Mattson--http://teacherlink.ed.usu.edu/tlnasa/reference/imaginedvd/files/imagine/YBA/HTCas-size/parallax3.html.

 

The parallax angle is half of the angle defined by drawing lines from Earth toward each of the two points where HT Cas appears in the images. Thus, the parallax angle is in the third dimension outside of the flat image. We can determine its value by comparing the movement of HT Cas to the distance between nearby stars whose angular separation is known. In this image, the angle between the two stars closest to HT Cas (Stars C and E in the previous question) is 0.000002778 degrees. You can use this angular distance between the two nearby stars as a "scale" to determine the angle of shift of the star HT Cas. For example, if the distance between the two nearby stars is 1" on your screen, and the distance between the position of HTCas in one season and where it will be in the next season is 2", then the angle of shift is 2 x 0.000002778 degrees (which will be twice the parallax angle). Notice that this is a very narrow angle--that's why the anicient Greeks could not see the parallax and so thought that the Earth could not be moving relative to the stars.

 

The Very First Measurement of the Distance to Another Star--Using the Original Data

Although the distance to our sun was estimated reasonably accurately more than 2200 years ago, the stars are so far away, and the angle of parallax so tiny, that the measurement of the distance to a star other than our sun did not happen until 1838 when Friedrich Bessel made a successful measurement of the parallax of the star 61 Cygni. He chose that star because it was near the North Star (and so was visible for most of the year--stars that appear in the southern sky of the northern hemisphere are only visible for part of the year), and because it was known to have a significant proper motion (astronomerese for the movement of a star not relatied to Earth's rotation). We are going to look at his original data and infer the distance to this star.

Bessel measured the angular distance to two nearby stars (of which we are only going to look at one). This angular distance changed through the year, and he successfully argued that this change was not due to random fluctuations (which can happen because the atmosphere can refract and bend light from the star and this can change from one night to another). He reported measurements taken over a little more than a year, from August 1837 to September 1838. Many nights were cloudy, of course, and so he does not have data for every single night. HIs angular distances are in seconds of arc, which are 1/3600 degree each.

Here are his results for the angular distance between 61 Cygni and the star that he called 'α.'

 

The Standard Candle Method

The parallax method only works for stars that are close enough for us to see their apparent shift in position as the Earth orbits the sun. This is not possible without telescopes, and even with telescopes, no one was able to measure the shift until Bessel did so in 1838. Even today, we can only measure the apparent shift in the closest stars and most stars in our own galaxy and all stars in other galaxies are much too far away for us to measure their distance by this method. Thus, we need a different method to measure the distance to farther-away objects. One other method is the standard candle method.

 

Henrietta Leavitt, Edwin Hubble, and the First Measurement of the Distance to the Andromeda Galaxy

Leavitt was an astronomer working at Harvard College Observatory to measure and catalog the brightness of thousands of stars. Some of those stars were variable stars, meaning that their brightness varied with time. Leavitt recognized that there was a pattern to the variation in brightness. How bright the stars appeared to be was correlated to how long it took the star to go through its cycle of getting dimmer and dimmer and then brighter and brighter. These stars have come to be known as Cepheid variable stars.

Leavitt was working with variable stars in the Magellanic Clouds--small galaxies just outside the MIlky Way that were much too far away to measure their distance by parallax. Thus, she knew their apparent brightness, but not their true brightness. However, she did know that all the stars had to be about the same distance away because they were all in a cluster in the Magellanic Cloud. Thus, the correlation between apparent brightness and cycle period meant that the true brightness must also be correlated to cycle period. In her own words from the report of her finding: "Since the variables are probably at nearly the same distance from the Earth, their periods are apparently associated with their actual emission of light, as determined by their mass, density, and surface brightness."

Here is the graph from the report of her findings.

The "Y" axis plots apparent magnitude of the stars--a measure of brightness where the bigger the number, the dimmer the star (thus 'up' on the graph corresponds to greater brightness. This is a logarithmic scale in which the brightness increase by x 2.512 for each decrease of magnitude of 1. So a change in magnitude from 16 to 12 corresponds to a change in brightness of nearly a factor of 40! A change of 5 magnitudes corresponds exactly to a change of x 100 in brightness. For comparision, the dimmest star one can see with the naked eye is about magnitude +6. Polaris, the North Star, a relatively dim star, is magnitude 1.97. Rigel, the brightest star in the constellation Orion, is magnitude 0.12. Sirius, the brightest star seen in the night sky, as magnitude -1.46. The full Moon is magnitude -12.74. The magnitude of the sun is -26.74. Clearly, all of the stars that Leavitt studied are much too dim to see without a telescope.

The "X" axis plots the log of the cycle period--how long it takes the star to go from its dimmest point to its brightest point and then back to its dimmest, measured in Earth days. So, a value of "1" corresponds to 10 days. A value of 2 corresponds to 100 days.

The top set of points and corresponding curve correspond to the maximum brightness for these variable stars, and the bottom line corresponds to the minimum brightness for these variable stars.

 

When Leavitt finished her work, the Cepheid variable were not yet standard candles because we did not yet know their true brightness, only how their brightness changed with cycle period. However, once Leavitt discovered this relationship, other researchers turned their telescopes on Cepheid variable stars that were much closer to Earth, close enough to allow us to figure out their distance using parallax. Leavitt's graph could then be recalibrated for absolute magnitude. (Absolute magnitude is the brightness that a star would have if were 10 parsecs away--a sort of standard used for true brightness by astronomers.)

In 1929, Edwin Hubble used Leavitts correlation between magnitude and Cepheid variable period, along with the calibrations to absolute magnitude available at the time, to measure the distance to what was then called the "great spiral in Andromeda." It was not known how far away this feature was, or even whether it was a feature within our galaxy or something outside our galaxy altogether. He measured the periods of many variable stars in the spiral in Andromeda, and showed that, without question, Andromeda was far outside the boundaries of the Milky Way Galaxy. The great implications of Hubble's work was not "wow, look how far away the nearest galaxy is" but rather, "wow, there are other places like our galaxy in the universe and they are really far away!"

We're going to reproduce a small portion of Hubble's work, using his data, although using modern correlations between the period of what are now called "classic" Cepheid variables and absolute magnitude.

 

Modern 'redos' of Hubble's measurements and calculations give the presently-favored distance of about 2.54 million light years to Andromeda. Edwin Hubbles calculations, using the best estimates for the calibration of Leavitt's correlation available at the time, was about 900,000 light years. Although smaller than the modern estimate, it still put the Andromeda "feature" far outside of the Milky Way Galaxy, and thus greatly expanded the size of the universe that we knew of. A large part of the difference between the modern measurements and Hubble's is the realization that not all Cepheid variables are "alike"--there are in fact two different groups that have different relationships between period and magnitude--not all Cepheid variables are 'classic' Cepheid variables.

 

For the mathematically inclined, you might also check out Aristarchus, Geometry, and the Discovery that Space is Big 

 

last updated 4/20//2020.  Thanks to NASA, ESI, S Beckwith, and the HUDF team for the Hubble Space Telescope Deep Field Image. thanks also to the NASA Goddard Spaceflight Center for images related to the HT Cas distance activity. Other text and pictures are the property of Russ Colson.