AST 114 – Spring 2002 Magnitudes and Stellar Brightness

Magnitudes and Stellar Brightness

What will you learn in this Lab?

For the first time we’ll be looking at the stars tonight as something more than just pretty points of light. We’re going to take special note of their location, their relative brightnesses, and even their colors. These types of observations were the fundamental work of both ancient and modern amateur astronomers alike – giving us a unique record of the brightnesses of stars through the course of history.

What do I need to bring to the Class with me to do this Lab?

For this lab you will need:

·  A copy of this lab script

·  A pencil

·  A scientific calculator

·  Your flashlight

·  Your field guide

·  Your star charts

·  Your star wheel


Introduction:

This exercise is designed to help the student become familiar with the magnitudes (i.e., apparent brightnesses) and colors of stars. While completing tonight's work, you should learn the naming convention used to identify stars.

Magnitudes: To describe how bright individual stars are and to express differences in brightness, astronomers assign each star a magnitude according to its brightness. The magnitude system was first used by the Greek astronomer Hipparchus, who divided the stars into six categories. He called the brightest stars in the sky `first magnitude' and the faintest `sixth magnitude'. Stars of intermediate brightness were assigned a number between 1 and 6. We still use a modified version of this system, with extensions to encompass a larger range of brightnesses by using numbers smaller than 1 and larger than 6.

The first thing you should notice about this system is that its ordering is backward: a small number (like 1, or even a negative number) denotes a brighter star than a large number (like 8 or 9). The brightest star in the night sky, Sirius, has a magnitude of -1.5 while the faintest star that can be seen with the unaided eye is about magnitude 6. For comparison, the magnitude of the Sun is -26.5. An easy way to remember that the magnitude system runs backward is to think of magnitudes in terms of rank. A `first-rank' star is brighter than a `second-rank' star, which in turn is brighter than a `third-rank' star, etc.

Another important feature of the magnitude system is that it is a logarithmic scale. When the system was first setup, it was thought that 1st magnitude stars were about twice as bright as 2nd magnitude stars, which were twice as bright as the 3rd magnitude stars, etc. Following this pattern, one would expect a 1st magnitude star to be about 32 (= 2 x 2 x 2 x 2 x 2) times brighter than a 6th magnitude star, but careful measurements have shown that this is not correct. Instead, the jump from 1st magnitude to 6th magnitude actually represents a change of 100 times in brightness. Each increase of one magnitude corresponds to a decrease by a factor of about 2.512 () in brightness. For example, how does the brightness of a star with magnitude 1 compare to that of a star with magnitude 4? It is 3 magnitudes brighter, which means that is 15.85 (=2.512 x 2.512 x 2.512) times as bright.

Finally, the basic magnitude system used here is one based on the apparent brightnesses of stars. That is, how they appear in the sky. Hence, we call these apparent magnitudes. Without knowing the distance to an individual star, we cannot say anything about its intrinsic brightness. A truly bright star that is very far away can appear to be fainter than an intrinsically dim star that is very close by.

Colors: Astute observers will note that the brighter stars have discernable colors.

While most stars look whitish to the eye, some have a distinctive blue, yellow, or reddish appearance. The perceived color depends primarily on the energy distribution of the star's emitted light, which is a function of the star's surface temperature. For blackbodies (and stars approximate blackbodies!), Wien's Law specifies the wavelength at which the peak intensity occurs. For high temperature stars this peak is at short wavelengths, while for cool stars the peak shifts to long wavelengths.

Another factor that influences the perceived color is the spectral response of the human eye, which is most sensitive to light in the green/yellow portion of the spectrum, diminishing at both blue and red wavelengths. Hence, hot stars will appear bluish to the eye, and cool stars will look reddish, provided they are sufficiently bright. Dim stars simply do not register enough light to trigger a color response from the eye, so the faint stars all appear white. Interested students should investigate the nature of rods and cones in the human retina.

Names of Stars: The brightest stars in the sky were given names in antiquity. Many of these, some from Arabic and Greek, are still used today (e.g., Sirius -- the brightest star in the sky). Fainter stars are sometimes named for the person who first noticed them (e.g., Barnard's star was found because of its very high proper motion across the sky). The majority of stars are simply designated by a catalogue number (e.g., BD +59º1915A is in the +59º declination zone of the Bonner Durchmusterung Catalogue).

The entire sky is divided into 88 constellations, so that every star is located in a constellation. Most of the brighter stars can be designated by a lower-case Greek letter and the (often abbreviated) constellation name (e.g., g Cas and b Ori). Usually the stars are named in order of brightness, with a being the brightest, b the next brightest, etc. Thus, the brightest star in Taurus (Aldebaran) is also called a Tau. Because this Greek letter/constellation name syntax is so common, you should become familiar with the lower-case Greek alphabet.

The Lower-case Greek Alphabet

a / alpha / i / iota / r / rho
b / beta / k / kappa / s / sigma
g / gamma / l / lambda / t / tau
d / delta / m / mu / u / upsilon
e / epsilon / n / nu / f / phi
z / zeta / x / xi / c / chi
h / eta / o / omicron / y / psi
q / theta / p / pi / w / omega

Inverse Square Law: this grand sounding title is given to the common sense phenomenon that the farther away you hold a light the fainter it appears to be. Since the light emitted from any light, including a star, spreads out in all directions, its intensity can be related to the surface area of a sphere. If you like, the light is being spread over the surface of a sphere: the further you get from the star, the larger the sphere must be, and so the greater the spreading, or fading, of the light intensity. This predicts that the further away you are from a star the fainter that star will appear. Now the equation for the surface area of a sphere is , where R is the radius of the sphere. So if you move double the distance away from a star, the surface area of the corresponding sphere quadruples. Similarly if you move three times farther away, the sphere’s surface area goes up nine times. But since the same amount of light is being spread over the surface of this sphere, the light intensity falls by proportion. This is why the relationship is called the inverse square law.

Stellar Distance and Magnitude: It stands to reason that if you know how bright a star appears to be and how bright the star really is, then you can figure out how far off that star should be. This raises the issue of the two main types of magnitudes that you will see referred to in Astronomy textbooks: Apparent and Absolute Magnitudes. The former of these is the regular observed magnitude we’ve been talking about already. The latter of these is something new. If you’re going to classify stars and describe them by how bright they are intrinsically, then you need a system by which to measure that intrinsic brightness. But how do you do that when the brightness of star depends on your distance from it? The way this is done is to define a standard distance from which to measure or define the standard brightness of star. This distance is 10 parsecs. The magnitude of a star measured at a distance of 10 parsecs is called the Absolute Magnitude and is the meter rule by which we classify stars of different types.

An equation relates the apparent and absolute magnitudes of a star (m and M respectively) to the distance to the star from the observer (d, measured in parsecs):

so if you know both m and M you can figure out the distance to the star.


Procedure

This exercise will give you the opportunity to attempt to replicate how stellar photometry (literally: the measurement of light) was done by civilizations such as the Ancient Chinese and Arabs. These peoples did their measurements totally by eye, and to this day this approach is still practiced by many amateur observers to keep track of variable stars as they change their apparent brightness in our night sky.

Choose a constellation, or two adjacent constellations, located high above the horizon, consisting of at least eight or more stars visible to your eye. Carefully draw this constellation, indicating the brightnesses of the stars in the same manner as they do on the star charts: with different sized dots, using larger dots for brighter stars and smaller ones for the fainter stars. Use your star charts to identify the stars you’ve drawn. In a table rank the stars in order of decreasing brightness, as they appear to you. (The brightest star should be first, the next brightest second, etc.). Record the star’s name and the constellation in which it resides. For each star make an estimate of its magnitude using your charts and/or the field guide. Use the legends on the charts to aid in your estimates. Also make an estimate of the star’s color.

Colors of stars indicate their surface temperatures. The hottest stars (T = 10000 to 25000 K) appear quite blue, while the coolest stars (T = 3000 to 5000 K) look reddish. Using these facts estimate which of the stars you’ve recorded is the hottest and which the coolest.

Now, return to looking at the sky and pick an area in the constellation you studied to examine more carefully. Pick your location well – it should have a good selection of bright and faint stars. Use a 40 mm eyepiece you have a nice wide angle view, and center the telescope on what you choose to be the best place. First, look through the finderscope, once the telescope is locked and driving, and make a quick finderchart to indicate where in the constellation you’re looking and label the main stars in the region, so you have a road map to your location. Make a detailed and proportional drawing of your field, as seen through the main telescope. Make sure your field has 2-3 stars of known brightness in it. Your TA will have AAVSO star charts for the region. These charts indicate stellar brightness in magnitudes for many stars. The table at the end of this lab script also lists the brightness for many of the main stars in the sky tonight.

Once you have a working diagram, start to estimate the brightnesses of the other stars in the field. There are a variety of ways to do this, but the one recommended here is to use the known stars to estimate how far between, or beyond, the brightness of the known stars your unknown star lies. Use the example on the next page as a guide to how you should proceed.

In this set of stars we have some brighter labeled stars, together with some unknown intermediate and faint stars. For this example let’s look at the star immediately between a, e and g. If we know that a has a magnitude of 2.3, g is 3.8 and e is 4.5, then we can make some estimate of how bright the unknown star is. This star is very close in brightness to e, but is slightly fainter. From looking at how much fainter than g e looks, and knowing that that corresponds to a magnitude drop of 0.7 magnitudes, we can estimate how much fainter the star in question is than e. I would estimate it to be about 0.1 magnitudes fainter, at most, giving it a magnitude of 4.6.

Again, let’s assume that we don’t know the brightness of d – but it appears to be somewhere between g and e in brightness. What estimate of its brightness can you make given the known values for g and e? This is how the process works, and how thousands of amateur astronomers around the world monitor the brightness of known variable stars to make their contribution to the world of astronomical research.

Use this technique to estimate the brightness of all the stars in your telescopic field. Also make some estimate of their color. Once you’ve done that, assume that all of the stars in your field have an absolute magnitude of +1.0. Calculate the corresponding distance to each star in parsecs based on your visual estimates of their apparent magnitudes.

Questions

Your lab report should address the following questions:

1.  From analysis of your tables of stellar brightness data, is it always true that the star labeled a is always the brightest, and b the second brightest, etc.? If not, why do you think that this might be so?

2.  How accurate do you think the measuring technique you used tonight really is? Why do you think this approach is better than just making a solitary measurement without comparing it to two or three stars at once? How reliable would you say amateur observations of variable stars are after having walked in their shoes?

3.  The magnitude system may seem an archaic one, but now you have seen how it was originally formulated and applied to stars in the sky. Given that the system is essentially logarithmic, how many times brighter in appearance is the brightest star we know of in the sky (Sirius with an apparent magnitude of 0.0) compared to a star down at the limit of our naked eye vision (at a magnitude of 6.0)? It is an astounding number and gives testament to how broad a range of light conditions our eyes can handle.