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The backbone of stellar astronomy is the classification of stars and a chart called the Hertzsprung–Russell Diagram (H-R Diagram). The reading that accompanies this lesson introduces a lot of new vocabulary and concepts related to the classification of stars. There is ample self-review in these online notes to help you with this.
After completing this lesson, you should be able to
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Before you go any further, ask yourself the question: "Do I really know what stars actually are?" If you are not at least 95 percent confident in your answer to the questions: "What are stars?" then take a few moments to visit NASA's Imagine the Universe, What is a Star?
In order to understand how we know what we do about the characteristics of the stars, we need to preview a couple of sections of the text from chapter 10: sections 10.1 and 10.2. Read those sections first and follow up with the next section where we define and work with the parallax of stars and the unit called a "parsec."
We observe parallax effects in our everyday activities. If you were to hold a pencil point to your computer screen, pointing at, for example, one of the bullets in a list or the period at the end of a sentence, and then move your head back and forth slightly, you would see the mark shift back and forth from the pencil point. Now, the pencil and the mark have not shifted in actuallity, only our perspective of them has. You perhaps noticed a parallax effect when riding a merry-go-round when you were young, watching your parents shift against the more distant rides as you approached and passed your parents by.
Let's take a look at the parallax of nearby stars as seen from the Earth orbiting the Sun. Here, the Earth represents you on the merry-go-round, the nearby star represents one of your parents, and the more distant stars the rest of the amusement park. Can you define "parallax" after viewing the animation?
The image above shows a "top view" of the Earth's orbit about the Sun and our view of a "nearby" star against a more distant star field. The image below shows how the star "moves" against the background stars over the course of the year for an observer located on Earth. The motion of the star here is greatly exaggerated!
Since the baseline is defined as 1 AU, the parallax angle is defined as 1/2 of the total shift (by convention, mostly).
The term "parsec" comes from an abbreviated form of
"parallax arc second." To understand parsec (abbreviated
pc) we need to recall what an arc second is. Chapter 1 of your
text briefly covered angles and what it means when we say that, for
example, two stars are 18 degrees apart in the sky. We can also talk
about the angular size of an object: the angular size of the Moon is 0.5
degrees. We can picture arcs across the sky; from horizon to horizon
(east-west, north-south, etc.) is 180 degrees. To the right is a sketch
of a protractor spanning 180 degrees. Think of the angular size of 1
degree on this protractor if you could stand in the middle of the bottom
edge; then, divide that 1 degree into 3600 slices. An arc second is
1/3600 of a degree, or very small indeed.
If we can measure the parallax angle of a star over the course of a year (or most recently, by the satellite Hipparchos) it is a trivial task to calculate the distance to the star, in parsecs. A star is 1 parsec away from the Sun if its parallax is 1 arc second; 2 parsecs away if its parallax is 0.5 arc seconds; 6 parsecs, 0.167 arc sec; and so on. The closest star to us, Proxima Centauri, has a parallax of about 0.7 arc seconds. So even the closest star has an extremely small shift against the background stars! Trying to measure these small shifts is difficult and prone to uncertainties, even with the most precise instruments. Plus, the background stars, against which astronomers are trying to gauge the shifts, also have parallaxes of their own.
Usually distances stated in light years are easier to understand. For example, since there are 3.26 light years (a distance measure) in a parsec, the distance to Betelgeuse in light years is 522 ly. We see Betelgeuse as it was 522 years ago, since it took light that long to "go the distance."
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How much of the Milky Way can we survey to get fairly accurate parallaxes for stars? The Hipparcos satellite observed parallaxes for about 2.5 million stars, out to a distance of about 1000 parsecs. The image, on the right, of the galaxy known as Messier 101 (M101, a galaxy the Milky Way is thought to resemble) indicates the diameter of the Galaxy in parsecs (red arrow) and the approximate area of the Milky Way (tiny red dot!) within which we can get measured parallaxes. As you will note, we don't get very far. Clearly another method must be used if we are to determine the extent of our galaxy, and one of these methods is discussed later and dealt with in the lab that accompanies this lesson.
Be sure to review Section 8.1 of your text as you read through these notes for a more complete discussion of the magnitude scale. Understanding magnitudes is fundamental to astronomy.
We will use the phrase "brightness of a star" to mean how "bright" a star looks as seen from Earth. Astronomers call this the star's apparent magnitude ("apparently that's how bright the star looks to us"). The brightness (apparent magnitude) of a star depends on
The authors of your text have downplayed the continued use of the magnitude scale, stating that it is outmoded and implying that it is seldom used by modern-day research astronomers. The magnitude scale may be outmoded, but it is certainly in very active use by today's astronomers, including those working on finding distances across the Universe! How do we go about finding the apparent magnitude of a star? We go to our observatory, open up the dome, point our telescope at a star, and record its light on a photometer—a meter that measures light. The photometer records how many photons we gathered from that star over a period of time. After some calculations that adjust the numbers for the efficiency of the detector, for the noise that is always present when using electronic detectors, and for the fact that even the night sky is not perfectly dark, we find the apparent magnitude of the star. There are catalogs of what astronomers call "standard stars," stars that have been observed many times and whose light output is fairly constant and well known. Our job is now much easier as all we have to do is measure the amount of light detected from an "unknown" star and measure the amount of light detected from a "known" star in the same part of the sky and at roughly the same time of night and compare the two.
Luminosity is the amount of power actually given off by the star (energy/sec or watts). A regular household lightbulb has a luminosity of 60 watts; the Sun's luminosity is approximately 4 × 1026 watts! The luminosity of a star depends on (but maybe not all at the same time, and in some ways that might be surprising)
Luminosity is an intrinsic property of the star.
For stars that are in the longest-lived part of their lives, where they are fusing hydrogen to helium in their cores (the so-called main sequence), there is a mass–luminosity relationship: the more massive the star, the more luminous it is. This fact will be explored in greater detail a bit later in the next lesson.
As if this system were not confusing enough already, it's going to get a bit worse. The problem facing stellar astronomers is that when we view the stars we are viewing a mixed bag of characters. Some are bright because they are close; some because even though they are very far away they are extremely luminous. Some stars are dim because they are low-mass and not very luminous, even though they may be close. Other stars are dim because they are thousands of parsecs away. How can we compare these stars and get down to their fundamental properties? We need to put the stars on a common scale. We can do this by putting their luminosities in terms of the Sun's luminosity, and this is what your text is showing in Fig. 9.13 when it introduces the H-R Diagram.
Observational astronomers use another scale, equivalent really to using the luminosity, called the absolute magnitude of a star. The absolute magnitude of a star is "the apparent magnitude it would have if it were located at 10 parsecs from the Sun." When we use the absolute magnitude of a star we
The Sun has an apparent magnitude of –26, but an absolute magnitude of 4.7. Recall that the smaller the number the brighter the star is; the absolute magnitude scale works the same way: the smaller the number the more luminous the star is. A distance star may have an apparent magnitude of 10.5, but an absolute magnitude of 2.5. Since its apparent magnitude is a larger number than its absolute magnitude, we know it is farther than 10 parsecs. Sirius, the brightest star in the sky, has an apparent magnitude of –1.5. Its absolute magnitude is 1.3. Since its absolute magnitude is a larger number than its apparent magnitude, we know it is closer than 10 parsecs. By convention, apparent magnitude is represented by m and absolute magnitude by M.
If we deal with the apparent magnitude of a star, then a difference of 1 corresponds to 2.512 times in brightness; a difference of 5 magnitudes corresponds to 100 times in brightness (2.512 * 2.512 * 2.512 * 2.512 * 2.512). If we deal with the absolute magnitude of a star, then a difference of 1 corresponds to 2.512 times in luminosity; a difference of 5 magnitudes corresponds to 100 times in luminosity (2.512 * 2.512 * 2.512 * 2.512 * 2.512).
The absolute magnitude can be calculated for any star as long as the star's distance d can be determined and we measure the star's apparent magnitude m at the telescope. While we often compare the characteristics of two stars, there is an extremely important formula that astronomers use where we work with the apparent magnitude and the distance to a star (determined by parallax measurements) to determine the absolute magnitude of the star. Recall, the absolute magnitude of a star is what is needed to put the stars on an "equal" basis for comparison. For right now, we will assume that we can find the star's distance by using its measured parallax. The text does not mention absolute magnitude, nor the magnitude equation, but this equation is of fundamental importance to an astronomer, and not that difficult to understand or work with.
M = m – 5 log (d) + 5
where M = absolute magnitude, m = apparent magnitude, and d = distance.
When discussing a difference in brightness or luminosity between two stars, you must deal with either the difference in the apparent magnitudes of the two stars or the difference in the absolute magnitudes of the two stars.
Consider the following data table for the 16 brightest stars in the sky (ignore the spectral type and luminosity class column for now):
| Common Name | Scientific Name | Apparent Magnitude |
Parallax (arc sec) | Absolute magnitude |
Spectral Class |
|---|---|---|---|---|---|
| Sun (Sol) | –26.72 | 4.83 | G2 V | ||
| Sirius | alpha Canis Majoris | –1.46 | 0.375 | 1.41 | A1 V |
| Canopus | alpha Carinae | –0.72 | 0.028 | –2.5 | A9 II |
| Arcturus | alpha Bootis | –0.04 | 0.090 | 0.2 | K1.5 III |
| Rigil Kentaurus | alpha Centauri | –0.01 | 0.751 | 4.4 | G2 V + K1 V |
| Vega | alpha Lyrae | 0.03 | 0.123 | 0.6 | A0 V |
| Capella | alpha Aurigae | 0.08 | 0.073 | 0.4 | G6 III + G2 III |
| Rigel | beta Orionis | 0.12 | 0.013 | –8.1 | B8 I |
| Procyon | alpha Canis Minoris | 0.38 | 0.288 | 2.6 | F5 IV |
| Achernar | alpha Eridanis | 0.46 | 0.026 | –1.3 | B3 V |
| Betelgeuse | alpha Orionis | 0.50 | 0.005 | –7.2 | M2 Iab |
| Hadar | beta Centauri | 0.61 | 0.009 | –4.4 | B1 III |
| Altair | alpha Aquilae | 0.77 | 0.198 | 2.3 | A7 V |
| Aldebaran | alpha Tauri | 0.85 | 0.048 | –0.3 | K5 III |
| Antares | alpha Scorpii | 0.96 | 0.024 | –5.2 | M1.5 Iab |
| Spica | alpha Virginis | 0.98 | 0.023 | –3.2 | B1 III + B2 V |
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Note: three of these stars are double or binary stars, as indicated by more than one spectral type in column 5 (the column you are supposed to ignore for now anyway) (Table adapted from Chris Dolan's Constellation and their Stars.) |
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Now let's take a look at the 16 closest stars in the sky.
| Common Name | Scientific Name | Parallax (arc sec) | Apparent Magnitude |
Absolute magnitude |
Spectral Class |
|---|---|---|---|---|---|
| Sun (Sol) | - | –26.72 | 4.83 | G2 V | |
| Proxima Centauri | V645 Centauri | 0.78 | 11.05 | 15.5 | M5.5 V |
| Rigil Kentaurus | alpha Centauri A | 0.75 | –0.01 | 4.4 | G2 V |
| alpha Centauri B | 0.75 | 1.33 | 5.7 | K1 V | |
| Barnard's Star | 0.54 | 9.54 | 13.2 | M3.8 V | |
| Wolf 359 | CN Leo | 0.42 | 13.53 | 16.7 | M5.8 V |
| BD +36 2147 | 0.40 | 7.50 | 10.5 | M2.1 V | |
| Luyten 726-8A | UV Ceti A | 0.39 | 12.52 | 15.5 | M5.6 V |
| Luyten 726-8B | UV Ceti B | 0.39 | 13.02 | 16.0 | M5.6 V |
| Sirius A | alpha Canis Majoris A | 0.38 | –1.46 | 1.4 | A1 V |
| Sirius B | alpha Canis Majoris B (white dwarf) |
0.38 | 8.3 | 11.2 | DA |
| Ross 154 | 0.35 | 10.45 | 13.1 | M3.6 V | |
| Ross 248 | 0.31 | 12.29 | 14.8 | M4.9 V | |
| Epsilon Eri | 0.30 | 3.73 | 6.1 | K2 V | |
| Ross 128 | 0.30 | 11.10 | 13.5 | M4.1 V | |
| 61 Cygni A | 0.29 | 5.2 | 7.6 | K3.5 V | |
| 61 Cygni B | 0.29 | 6.03 | 8.4 | K4.7 V | |
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These stars are ranked in order of their distances, but we can ask similar questions to those given for the brightest stars:
Take a look also at Appendices 10 and 11 of your text for a slightly different look at these same stars, including the different names they may have.
Let's continue here in our efforts to categorize and classify these stars to try to make some sense of their characteristics. Astronomers classify stars
This system of using letters dates back to the turn of the century and the pioneering work of Annie Jump Cannon. Initially, the stars were assigned their types in order: A, B, C, etc., primarily based upon the strength of the hydrogen lines in the spectra. This was well before astronomers understood how the lines in stellar spectra were actually physically produced in the stars. Eventually, it was realized that O stars and B stars had weak hydrogen lines not because they were cool, but rather because they were hot—so hot that the hydrogen atoms were being ionized. Many of the principle letters were dropped, giving us the spectral sequence: O B A F G K M. Remember, this sequence reflects the temperature of the stars, not their compositions. Stars are all composed of hydrogen (roughly 75%) and helium (roughly 25%) with just a trace of additional elements. Even though some stars have slightly more of these trace elements than other stars, the effects on their spectra are extremely subtle.
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After giving stars their main spectral type, it became obvious, as more and more stars were classified and finer details of their spectra were included, that just a principle letter wouldn't do; subclasses were needed. The subclasses run from 0 to 9 for most stars. For example, F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 G0 etc., with F0 being the hottest of the F stars and F9 being the coolest. An F9 star would be slightly hotter than a G0 star. In practice, the spectral classifications range from O3 to M8, with finer delineation used: F1.5, K2.7, etc. (First, we make it complicated, then we simplify it, and then we make it more complicated!)
You should work with Section 8.3 and Table 8.1 in your text, and the image on the right to get a sense of how the stars are classified.
Astronomers also classify stars by their luminosity, putting them into luminosity classes using Roman numerals as the identifiers (see also Figures 9.13 and 9.14 in your text): main sequence (V), subgiant (IV), giant (III), bright giant or dim supergiant (II), and supergiant (I). It takes a discriminating eye to tell the differences between luminosity classes for a given spectral type. Take a look at the effects of luminosity for spectral type A0 stars:
The spectra show that the more luminous the star, for a given spectral type, the narrower the lines are. The lines identified are those from hydrogen atoms (H plus a Greek letter) and calcium atoms that have lost one electron (CaII).
Let's put this all together by looking at a sketch of an H-R Diagram. This particular plot is useful in that you can see the relationship between absolute magnitude and luminosity and spectral type and temperature. The location of only a few stars is indicated to emphasize the regions of the H-R Diagram where stars are found.
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Discovered simultaneously by two separate astronomers, the H-R Diagram became the foundation for our understanding of the stars. We will investigate the physical processes in stars that lead to their positions on the H-R Diagram in the next lesson. For now, let's compare where the brightest stars are located on the H-R Diagram compared to the closest stars. Linked here is an H-R Diagram of the 26 brightest and 26 of the nearest stars.
Study the diagram and see if you can answer the questions in the self-study exercise that follows; these questions are important for your understanding of not only our neighborhood but also the stellar population of the Galaxy. The answers are given, but try first to answer the questions by using hints given in your text in Sections 9.1 and 9.4.
It seems to be a reasonable to assume that stars positioned at close to the same place on the H-R Diagram are also reasonably close in their intrinsic characteristics. We must also assume that we have been reasonably accurate in determining their spectral types (or temperatures) and absolute magnitudes (or luminosities). If we then have a large sample of stars for which we have an independent measure of their distances, say by the method of measured parallax, we can use this sample of stars and their positions on the H-R Diagram to derive the distances of stars that are way too far to get a measured parallax. With today's large telescopes, we are capable of observing the spectra of extremely distant stars across the Milky Way Galaxy and in nearby galaxies. The method that uses the spectrum of a star to determine its distance is given the slightly misleading name of "spectroscopic parallax." Review Section 10.4 of your text in connection with the following discussion and demonstration.
The method has nothing to do with measuring any parallaxes. In that sense, it is a bit of a misnomer. But we do use the spectrum of a star (and remember, we can usually always get a spectrum of a star, as long as it's not too far away). Knowing the spectral type and luminosity class of a star, we can use our knowledge of how stars are positioned on the HR Diagram (those whose distances we do know) to derive the absolute magnitude of the star whose distance we want to find. We do this by matching the spectral type with the luminosity class region on the H-R Diagram. Once we have the star's absolute magnitude and the apparent magnitude (which we can always measure, as long as the star is not too far away), then we can use what we are calling the magnitude equation, shown below. The distance to a star is equal to 10 raised to the power of (m – M + 5)/5. (The lab will refresh your memory as to how to work with exponents.)
Here is an animation that demonstrates how to determine the absolute magnitude of a star, once you know the star's spectral type and luminosity class.
| Star | Measured Parallax (arc sec) |
Distance (d) in Parsecs |
Answers |
|---|---|---|---|
| Sirius | 0.37 | ||
| Vega | 0.13 | ||
| Antares | 0.008 | ||
| Fomalhaut | 0.145 | ||
| Betelgeuse | 0.00625 |
| (Imaginary) Star Name |
Apparent Magnitude (m) |
Absolute Magnitude (M) |
Closer? | Farther? |
|---|---|---|---|---|
| Abba | 1.5 | –2.0 | ||
| Dabba | –1.5 | 2.0 | ||
| Gabba | 12.5 | 12.5 | ||
| Zabba | 2.5 | 2.0 | ||
| Star | Apparent Magnitude (m) |
Distance (d) in Parsecs |
Absolute Magnitude (M) |
|---|---|---|---|
| Sirius | –1.5 | 2.7 | |
| Vega | 0.0 | 7.7 | |
| Antares | 0.9 | 125 | |
| Fomalhaut | 1.15 | 6.9 | |
| Betelgeuse | 0.4 | 160 |
| Star Name | Spectral Type | Rank |
| Barnard's Star | M5 | |
| alpha Centauri A | G2 | |
| Sirius A | A1 | |
| 61 Cygni A | K5 | |
| tau Ceti | G8 | |
| Kapteyn's Star | M0 | |
| epsilon Eridani | K2 | |
| Procyon | F5 | |
| beta Centauri | B1 | |
| Sun | G2 |
| giant | I | |
| main sequence | II | |
| bright giant | III | |
| subgiant | IV | |
| supergiant | V |
| H-R Diagram | Star | Spectral Type |
Luminosity Class |
Absolute Magnitude |
|---|---|---|---|---|
|
Arcturus | K2 | III | |
| alpha Centauri | G2 | V | ||
| Vega | A0 | V | ||
| Betelgeuse | M2 | I | ||
| Altair | A7 | V | ||
| Aldebaran | K5 | III | ||
| Regulus | B7 | V | ||
| Sun | G2 | V | ||
| Note: the " ~ " mark that you see with the answers means "approximately," emphasizing, again, that these are only good estimates. | ||||
We'll start this part with a short review of what we've been able to determine about stars from our observations of magnitudes, luminosities, parallaxes, spectra, radii. We'll find that the luminosities of stars have a HUGE range in values -- from 10-3 times that of the Sun to over 106 times that of the Sun! Over a range of 9 powers of 10. The radii of stars go from 1/1000 times the Sun to about 1000 times the radius of the Sun, 6 powers of 10. Temperatures of stars, on the other hand, range from a low of about 3000 K to maybe 60,000 K, or only a factor of 20. Why the differences? Aren't we missing something?
Stars like company, close company. Over 50% of the stars (some estimates say closer to 70% of the stars) in the sky are part of a binary or multiple star system. How do we know this? How do we find out whether or not two stars that look close together in the sky are actually gravitationally bound? Can some stars be so close together that even the most powerful telescopes cannot resolve them?
OR
OR
We also classify stars by their evolutionary stage and their luminosity class.
That is, main sequence (V) (dwarf), subgiant (IV), giant (III), bright giant or
dim supergiant (II), and supergiant (I).
Spectral Types: O B A F G K M
Then subclasses: For example F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 G0 etc.
The spectral classifications range from O3 to M8.
Something is definitely missing. Let's say we are describing someone in our class. We know where she is sitting, we know the color of her clothes, how far away she is from her friends, her body temperature. But, we don't have the full story! How tall is she? How much does she weigh? (Her age will have to wait for the next chapter.)
We could never know the exact mass of a star if it were not for binaries. At least 50 per cent of the stars in the sky (and probably even more) are members of binary or multiple star systems. To determine masses, we must rely on our old friend, Newton's version of Kepler's third law:
but, if we express the period in years, the semi-major axes in AU's, and the masses in terms of our Sun's mass, then we can simplify it tremendously:
We now have the sum of the masses. We have, however, only one equation with two unknowns. We assume we can measure the distance each star is from the center of mass of the system, so that we know a1 and a2. We can call in another simple principle of physics: the teeter-totter principle: If the two masses are balanced, the center of mass is in the center. Substitute more massive objects on one end, and the center of mass gets closer and closer to the more msasive object. For a teeter-totter, that center of mass was where the balance point was. [Recall also that a very heavy person would hardly move up and down at all, while the very light person on the other end got the ride of his/her day!]
The symbols a1 and a2 represent the distances
of each star to the center of mass of the system. We now have two equations with
two unknowns.
Wait a minute? How did we measure the distances? (Think: parallax!)
Before you continue with your study of visual binaries, take some time to review just how orbits work. What happens when you vary the eccentricities of orbits? the masses of the stars? the distances they are apart? What does it mean when we talk about "center of mass"? Start with this simple example of Kepler's Third Law and watch how the period changes as you vary the mass of the primary star and the distance of the secondary star.
We can observe visual binaries over a number of years and watch them as
they orbit the center of mass. We can then determine the angular
separations. Then, we use something very powerful, the small-angle formula.
When angles are extremely small, then the sine and tangent of the angle are
approximately equal to the angle itself. (If you don't believe this, put in a
small angle, in radians, in your calculator and take the sine and tangent of
it.) Using the small angle formula, we determine the distance to the star, we
measure the angular size of the separation, and then use the formula:
We can also calculate the semimajor axis in those cases where we can measure the
orbital velocity, v. We can, without too much pain, assume the orbit
is a circle of radius a, making the circumference of the orbit
2*pi*a. We have measured the period, p. We know that the
velocity of the orbit is equal to the distance traveled in one orbit divided by
the time it took it to orbit (p). That's 2*pi*a/p.
Then, simply solve for a:
[semi-major axis] =
[period*velocity]/[2*pi].
What if the stars are too close together or too far away to see the individual
stars? Then we must rely on another procedure to unravel the system. We go to
our telescopes with their spectrographs and obtain spectra of the stars. We
monitor the stars over time and watch the absorption lines shift from blue to
red (red to blue). We get the motion along the line of sight for the two stars.
We must be able to see the absorption lines from both of the stars
(double-lined spectroscopic binary) for this to work. By watching the relative
shifts of the two sets of absorption lines, we get an idea of the ratios of the
masses of the two stars -- the relative velocities of the two stars
around their center of mass are inversely proportional to their
relative masses. Check out this
simulation of a spectroscopic binary and confirm this for yourself.
The Doppler effect gives motion only along our line of sight (radial motion).
We really cannot know the exact inclination angle unless the binary system
eclipses. Again, to see for yourself, work a few parameters using the
Eclipsing Binaries simulation.
The light curve from eclipsing binaries also gives us information about the radii of the two stars. We know the velocity of the stars, and we know how long the eclipses last: # of kilometers/second * # of seconds = # of kilometers. Astronomers have also been able to measure the angular diameters of a few 10's of stars that lie along the ecliptic by timing occultations of the stars by the Moon. A technique known as "speckle interferometry" (look it up!) has also yielded a few more angular diameters. Of all the stars, only Betelgeuse is large enough for us to actually see the disk of the star.
From the text:
