How Bright are the Stars?

171-172

Now that we know something about distances to the stars we can answer an even more subtle question - how bright are the stars? There is obviously more than meets the eye in this "child-like" question as the following example reveals.

Figure 8.2 The Summer Triangle

Figure 8.2 shows the "summer triangle" - a lovely asterism that graces Canadian skies in the summer and early fall. The three bright stars - Vega, Altair and Deneb are the "first stars" of the evening and form an easily seen triangle (hence the name). A closer look at the summer triangle reveals something quite remarkable.

• Consider the following "data":
• Vega: magnitude = 0.00, parallax = 0.129"
• Altair: magnitude = 0.75, parallax = 0.194"
• Deneb: magnitude = 1.25, parallax = 0.001

Given this data, which of the three stars is the "brightest"?

Before you can answer this question, a more precise meaning will need to attached to the term "brightness".

Brightness and Flux

The summer triangle shows us that while Deneb is almost 200 times farther away from us than Altair, both stars appear to be similar in brightness. Deneb must, in some fundamental sense be much brighter than Altair. At its simplest, brightness refers to the amount of energy (in the form of photons) entering your eye or hitting the detector of your digital camera. When you measure brightness without any consideration of the affect of distance then you are using apparent brightness.

So, how does distance affect brightness? Try the following applet which shows a sphere illuminated from the centre by a bright light source. As you adjust the radius of the sphere (use the scrollbar on the bottom of the applet) you will see that the light is distributed over the surface of a sphere and that the amount of light per square meter drops rapidly as the sphere grows.

Astronomers call the amount of energy per square meter that falls on a surface Flux. When you observe a star, the brightness that you perceive is related to the flux received from the star. Use the applet to investigate how flux and distance are related.

Example 8.5 How would the energy received from an object change if the object were moved twice as far away?

Solution: Measure the flux at a point where the radius of the sphere in the applet is 2.0 m (this is shown numerically on the applet). The applet should report that the flux is 25 W/m ² . Now increase the radius to 4.0 m. The applet now reports a flux of 6.25 W/m ². If you express these numbers as a ratio you will see that the flux decreases by a factor of 4 when you increase the distance from the source by a factor of 2. This is called an inverse-square relationship. So, doubling the distance from an object means that we will receive one quarter as much energy from that object.

We can now make an important distinction: Apparent brightness is related to the amount of energy we receive from an object while intrinsic brightness is related to the actual amount of energy being emitted by an object.

Because the flux from a star depends in a rather simple way on distance from the star it is relatively easy to account the affect that distance will have on the apparent brightness of the star. This idea will be explored in the next section.

Example 8.6 Estimate how much intrinsically brighter Deneb is than Altair.

Solution: By comparing the parallaxes of the stars we infer that Deneb is almost 200 times farther away than Altair. Make the simplifying assumption that they are the same apparent magnitude. Since flux follows an inverse-square relation an increase in distance by 200 times would result in a decrease in brightness by (1/200)² = 1/40,000 times. Since Deneb appears to about the same brightness Altair we conclude its intrinsic brightness must be about 40 thousand times as bright as Altair. Deneb is one of the intrinsically brightest stars visible to the un-aided eye. (Note - if we take into consideration the fact that Altair is one-half a magnitude brighter visually than Deneb our estimated comparison in intrinsic brightness is revised downward to a factor of 25 thousand.)

Getting a Level Playing Field: The Absolute Magnitude Scale

If the stars were all the same distance from us then comparing magnitudes would be the same as comparing brightness. To "remove" the affect of distance on brightness astronomers have developed a different magnitude scale - the Absolute Magnitude. The absolute visual magnitude is written M_V and is defined as the visual brightness a star would have if the star was located 10 parsecs away from you. In contrast, the apparent visual magnitude, denoted m_V is the magnitude measure you have used thus far in the course and is a measure of what you see. Absolute magnitude gives you a much better idea of the intrinsic brightness of a star and ultimately is a measure of how much energy is being emitted by the star.

As you just saw, distance affects brightness in a manner that can be understood mathematically and this provides us with a mathematical way to relate apparent magnitude, absolute magnitude and distance. This relationship is so important that astronomers call the difference between apparent and absolute magnitudes the distance modulus. Distance modulus is expressed via the equation:

Distance modulus is a proxy for distance. If a star's distance modulus is greater than zero then the star is more than 10 pcs away. If, on the other hand the distance modulus is negative then the star is closer to us than 10 pcs.

Example 8.7 Use the data given below to rank order the following stars from closest to farthest way.

Solution: Determine the distance modulus for each star and then rank order from smallest distance modulus to greatest. Since distance modulus is a proxy for distance this will also tell you the distance ranking.

In this example the distance order from nearest to farthest is: Sirius, Vega, Betelgeuse, Polaris.

An Interactive Distance Modulus Graphing Calculator:

The applet given below will help you further explore the way in which distance modulus and distance to a star are related. You will need this applet to help answer the following questions as well as questions in future parts of the course.

To understand how much energy stars emit

Chapter 9.2

Flux is the measure of the amount of energy per square meter falling on a surface. The units for flux are W/m ².

The absolute magnitude of a star is defined as the magnitude a star would have if that star was located 10 parsecs away from you

Example 8.8 Use the data given in Example 8.7 to estimate the distance to the north star - Polaris.

Solution: You will need the distance modulus to answer this. From the data given m_v-M_v= 5.61. If you read directly from the graph you will first need to change the scale by clicking on the radio buttons in the lower left corner. Next, select the graph and drag the inspection tool along the graph to get an approximate distance of 1.3 pc. If you want, however, you can get a more precise value by entering the appropriate numbers into one of the two numerical calculators to get distance = 132 pcs.

As a final example in this section consider the following more challenging question.

Example 8.9 You observe two stars - one has an apparent magnitude of 1.2 and a parallax of 0.1" while the other has m = 3.2 and p = 0.035". Which of the two is intrinsically brightest? 

Solution: If we knew the distance modulus for each star we could then determine the absolute magnitude of each star. This would then allow us to answer the question - which is intrinsically brightest. To start:

1. Use the parallax information to determine the distance of each star.
2. Next, use the graphing calculator to determine the distance modulus for each star.
3. Calculate the absolute magnitude for each star

Distance: Use parallax! Summarize in the following table:

Use the distance modulus calculator to determine the distance modulus for each star:

Now, re-arrange the distance modulus formula to read and insert numbers to get:

Star "B" is intrinsically brighter since it has a lower absolute magnitude value. (Remember - the magnitude scale goes backwards!!)

Luminosity and Flux

To finally understand how much energy a star emits you will need one more concept - luminosity. The idea of luminosity combines the concept of Flux (energy a star emits per square meter per second) with the geometric idea of surface area. When these two ideas are combined by multiplying, you have the total amount of energy emitted each second - or luminosity. This is defined as:

Luminosity = (Flux) X (Surface Area)

Example 8.10 Imagine (as you will learn in later chapters) that as a star ages an cools, its surface flux drops by a half and at the same time its size doubles. How will that change the star's intrinsic brightness?

Solution: The key here is to remember how sizes and areas relate to each other. In the case of a sphere (which you are assuming the star is) the area varies with the square of the radius. So doubling the size of the star will quadruple its area. Therefore, to determine how the intrinsic brightness will change find the luminosity. Since we have half the original flux but 4 times the area, we conclude that:

New Luminosity = (1/2)Flux X( 4)Area = 2 X Old Luminosity

The star is now twice as bright (intrinsically) than before.

Practice

1. Sirius has an apparent magnitude of -1.45 and Rigel has an apparent magnitude of 0.15. Why can't you conclude which of the tow stars is intrinsically brightest using only the information given? What additional information do you need?
2. Two stars have the same surface temperature but one star is 3 magnitudes brighter than the other. Give two possible explanations for the difference in brightness.
3. In a future chapter you will learn about neutron stars. Neutron stars have surfaces that are millions of degrees K and yet are exceedingly faint. Why do you suppose they are so faint?
4. An A0V star has an apparent magnitude of 8. If you want to use the method of spectroscopic parallax to determine the distance to this star what additional piece of information will you need to get?
5. Complete question 4 above - find the missing information and use spectroscopic parallax to determine the distance. What possible factors could make your estimate unreliable?