# Approximate spectral type and luminosity given apparent magnitude and distance

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I am looking at the data acquired from the Gaia DR2 survey. I found that most of the stars had their distance and apparent magnitude catalogued, but not their spectral type and luminosity class. Is there a way to approximate this?

You can calculate the absolute magnitude of a star: $$M=m-5log_{10}(frac{d}{10, ext{pc}})$$ where $$M$$ is absolute magnitude, $$m$$ is apparent magnitude and $$d$$ is the distance.

Then you take a look at the HR diagram.

One can easily see, that you need two data to obtain the third one, but we have only one data (absolute magnitude). That means, that you also have to know the luminosity classification of a star:

• I: supergiants
• II: bright giants
• III: giants
• IV: subgiants
• V: main sequence
• VI: white dwarfs

Then you just look at the intersection of the class and absolute magnitude.

If you know radius, then this is even easier. All you need is this formula: $$T={Big{(}frac{10^{0.4(4.77-M)}}{4pi R^2 sigma}Big{)}}^{0.25}$$ $$T$$ is the temperature in Kelvins, $$M$$ is absolute magnitude of the star, $$R$$ is radius of the star, $$sigma$$ is Stefan-Boltzmann constant.

With these data you look at the table of temperature and spectral classes or HR diagram.

If you are given color, then you already have an answer (spectral class is defined with color). If you are given temperature, you can look at the HR diagram.

## Apparent magnitude

Apparent magnitude ( m ) is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's light caused by interstellar dust along the line of sight to the observer.

The word magnitude in astronomy, unless stated otherwise, usually refers to a celestial object's apparent magnitude. The magnitude scale dates back to the ancient astronomer Ptolemy, whose star catalog listed stars from 1st magnitude (brightest) to 6th magnitude (dimmest). The modern scale was mathematically defined in a way to closely match this historical system.

The scale is reverse logarithmic: the brighter an object is, the lower its magnitude number. A difference of 1.0 in magnitude corresponds to a brightness ratio of 5 √ 100 , or about 2.512. For example, a star of magnitude 2.0 is 2.512 times brighter than a star of magnitude 3.0, 6.31 times brighter than a star of magnitude 4.0, and 100 times brighter than one of magnitude 7.0.

The brightest astronomical objects have negative apparent magnitudes: for example, Venus at −4.2 or Sirius at −1.46. The faintest stars visible with the naked eye on the darkest night have apparent magnitudes of about +6.5, though this varies depending on a person's eyesight and with altitude and atmospheric conditions. [1] The apparent magnitudes of known objects range from the Sun at −26.7 to objects in deep Hubble Space Telescope images of magnitude +31.5. [2]

The measurement of apparent magnitude is called photometry. Photometric measurements are made in the ultraviolet, visible, or infrared wavelength bands using standard passband filters belonging to photometric systems such as the UBV system or the Strömgren uvbyβ system.

Absolute magnitude is a measure of the intrinsic luminosity of a celestial object rather than its apparent brightness and is expressed on the same reverse logarithmic scale. Absolute magnitude is defined as the apparent magnitude that a star or object would have if it were observed from a distance of 10 parsecs (3.1 × 10 14 kilometres). When referring to just "magnitude", apparent magnitude rather than absolute magnitude is normally intended.

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The traditional name Bellatrix is from the Latin bellātrix "female warrior" it first appeared in the works of Abu Ma'shar al-Balkhi and Johannes Hispalensis, where it originally referred to Capella, but was transferred to Gamma Orionis by the Vienna school of astronomers in the 15th century, and appeared in contemporary reprints of the Alfonsine tables. [13] In 2016, the International Astronomical Union organized a Working Group on Star Names (WGSN) [14] to catalog and standardize proper names for stars. The WGSN's first bulletin of July 2016 [15] included a table of the first two batches of names approved by the WGSN which included Bellatrix for this star. It is now so entered in the IAU Catalog of Star Names. [16] The designation of Bellatrix as γ Orionis (Latinized to Gamma Orionis) was made by Johann Bayer in 1603. The "gamma" designation is commonly given to the third-brightest star in each constellation

Bellatrix is a massive star with about 7.7 times the Sun's mass. It has an estimated age of approximately 25 million years—old enough for a star of this mass to consume the hydrogen at its core and begin to evolve away from the main sequence into a giant star. [17] The effective temperature of the outer envelope of this star is 22 000 K , [8] which is considerably hotter than the 5,778 K on the Sun. This high temperature gives this star the blue-white hue that occurs with B-type stars. [18] The measured angular diameter of this star, after correction for limb darkening, is 0.72 ± 0.04 mas. [19] At an estimated distance of 250 light-years (77 parsecs), [1] this yields a physical size of about six times the radius of the Sun. [20] [17]

Bellatrix was thought to belong to the Orion OB1 Association of stars that share a common motion through space, along with the stars of Orion's Belt: Alnitak (Zeta Orionis), Alnilam (Epsilon Orionis), and Mintaka (Delta Orionis). However, this is no longer believed to be the case, as Bellatrix is now known to be much closer than the rest of the group. [17] It is not known to have a stellar companion, [21] although researchers Maria-Fernanda Nieva and Norbert Przybilla raised the possibility it might be a spectroscopic binary. [22] A 2011 search for nearby companions failed to conclusively find any objects that share a proper motion with Bellatrix. Three nearby candidates were all found to be background stars. [23]

Some reseachers thought that Bellatrix was a member of 32 Orionis group. The 32 Ori group should in fact be termed the Bellatrix Cluster on the basis that the sky position, distance of Bellatrix are similar to those of the 32 Ori group. [24]

Bellatrix has been used as both a photometric and spectral standard star, but both characteristics have been shown to be unreliable.

In 1963, Bellatrix was included with a set of bright stars used to define the UBV magnitude system. These are used for comparison with other stars to check for variability, and so by definition, the apparent magnitude of Bellatrix was set to 1.64. [25] However, when an all-sky photometry survey was carried out in 1988, this star was itself found to be variable. It ranges in apparent magnitude from 1.59 to 1.64. [26]

The spectral types for O and early B stars were defined more rigorously in 1971 and Bellatrix was used as a standard for the B2 III type. [27] The expected brightness of Bellatrix from this spectral type is about one magnitude brighter than calculated from its apparent magnitude and Hipparcos distance. [28] Analysis of the observed characteristics of the star indicate that it should be a B2 main sequence star, not the giant that it appears from its spectral type. [29] Close analysis of high resolution spectra suggest that it is a spectroscopic binary composed of two similar stars less luminous than a B2 giant. [22]

Bellatrix was also called the Amazon Star, which Richard Hinckley Allen proposed came from a loose translation of the Arabic name Al Najīd, the Conqueror. [10] A c.1275 Arabic celestial globe records the name as المرزم "the lion". [30] Bellatrix is one of the four navigational stars in Orion that are used for celestial navigation. [31]

In the 17th century catalogue of stars in the Calendarium of Al Achsasi al Mouakket, this star was designated Menkib al Jauza al Aisr, which was translated into Latin as Humerus Sinister Gigantis. [32]

The Wardaman people of northern Australia know Bellatrix as Banjan, the sparkling pigment used in ceremonies conducted by Rigel the Red Kangaroo Leader in a songline when Orion is high in the sky. The other stars of Orion are his ceremonial tools and entourage. Betelgeuse is Ya-jungin "Owl Eyes Flicking", watching the ceremonies. [33]

To the Inuit, the appearance of Betelgeuse and Bellatrix high in the southern sky after sunset marked the beginning of spring and lengthening days in late February and early March. The two stars were known as Akuttujuuk "those (two) placed far apart", referring to the distance between them, mainly to people from North Baffin Island and Melville Peninsula. [34]

## Project Structure

The first two sections of the project give a basic introduction to H-R diagrams. They also illustrate the differences between the brightest stars we see in the night sky and the closest stars to our Sun. These two sections can be done on their own as a short lesson for a lower-level class, or if time does not permit a deeper exploration of the topic.

The next section gets into the difficulties of determining the distances to stars. It uses data from the Hipparcos satellite. Students learn how to calculate the distances to stars using parallax. They will use the distances to find absolute magnitudes and to create an H-R diagram for a star cluster.

There is an optional section on calculating the radius of a star. This problem involves a lot of math, but should be accessible to a strong Algebra II student. Not doing this section will not diminish the material on H-R diagrams, but some students may find it a rewarding challenge to find a fundamental property of other stars.

Globular clusters are very far away. You can assume that all the stars are at the same distance. Students will see the difficulty of creating an H-R diagram due to the large amount of data necessary. They will then use a simple tool to search the data and make and H-R diagram for a globular cluster using hundreds of data points.

## Collaborative Group Activities

What two factors determine how bright a star appears to be in the sky?

Explain how parallax measurements can be used to determine distances to stars. Why can we not make accurate measurements of parallax beyond a certain distance?

What would be the advantage of making parallax measurements from Pluto rather than from Earth? Would there be a disadvantage?

Parallaxes are measured in fractions of an arcsecond. One arcsecond equals 1/60 arcmin an arcminute is, in turn, 1/60th of a degree (°). To get some idea of how big 1° is, go outside at night and find the Big Dipper. The two pointer stars at the ends of the bowl are 5.5° apart. The two stars across the top of the bowl are 10° apart. (Ten degrees is also about the width of your fist when held at arm’s length and projected against the sky.) Mizar, the second star from the end of the Big Dipper’s handle, appears double. The fainter star, Alcor, is about 12 arcmin from Mizar. For comparison, the diameter of the full moon is about 30 arcmin. The belt of Orion is about 3° long. Keeping all this in mind, why did it take until 1838 to make parallax measurements for even the nearest stars?

The Sun is much closer to Earth than are the nearest stars, yet it is not possible to measure accurately the diurnal parallax of the Sun relative to the stars by measuring its position relative to background objects in the sky directly. Explain why.

Explain why color is a measure of a star’s temperature.

What is the main reason that the spectra of all stars are not identical? Explain.

What elements are stars mostly made of? How do we know this?

What did Annie Cannon contribute to the understanding of stellar spectra?

Name five characteristics of a star that can be determined by measuring its spectrum. Explain how you would use a spectrum to determine these characteristics.

How do objects of spectral types L, T, and Y differ from those of the other spectral types?

Do stars that look brighter in the sky have larger or smaller magnitudes than fainter stars?

The star Antares has an apparent magnitude of 1.0, whereas the star Procyon has an apparent magnitude of 0.4. Which star appears brighter in the sky?

Based on their colors, which of the following stars is hottest? Which is coolest? Archenar (blue), Betelgeuse (red), Capella (yellow).

Order the seven basic spectral types from hottest to coldest.

What is the defining difference between a brown dwarf and a true star?

### Thought Questions

If the star Sirius emits 23 times more energy than the Sun, why does the Sun appear brighter in the sky?

How would two stars of equal luminosity—one blue and the other red—appear in an image taken through a filter that passes mainly blue light? How would their appearance change in an image taken through a filter that transmits mainly red light?

[link] lists the temperature ranges that correspond to the different spectral types. What part of the star do these temperatures refer to? Why?

Suppose you are given the task of measuring the colors of the brightest stars, listed in Appendix J, through three filters: the first transmits blue light, the second transmits yellow light, and the third transmits red light. If you observe the star Vega, it will appear equally bright through each of the three filters. Which stars will appear brighter through the blue filter than through the red filter? Which stars will appear brighter through the red filter? Which star is likely to have colors most nearly like those of Vega?

Star X has lines of ionized helium in its spectrum, and star Y has bands of titanium oxide. Which is hotter? Why? The spectrum of star Z shows lines of ionized helium and also molecular bands of titanium oxide. What is strange about this spectrum? Can you suggest an explanation?

The spectrum of the Sun has hundreds of strong lines of nonionized iron but only a few, very weak lines of helium. A star of spectral type B has very strong lines of helium but very weak iron lines. Do these differences mean that the Sun contains more iron and less helium than the B star? Explain.

What are the approximate spectral classes of stars with the following characteristics?

1. Balmer lines of hydrogen are very strong some lines of ionized metals are present.
2. The strongest lines are those of ionized helium.
3. Lines of ionized calcium are the strongest in the spectrum hydrogen lines show only moderate strength lines of neutral and metals are present.
4. The strongest lines are those of neutral metals and bands of titanium oxide.

Look at the chemical elements in Appendix K. Can you identify any relationship between the abundance of an element and its atomic weight? Are there any obvious exceptions to this relationship?

Appendix I lists some of the nearest stars. Are most of these stars hotter or cooler than the Sun? Do any of them emit more energy than the Sun? If so, which ones?

Appendix J lists the stars that appear brightest in our sky. Are most of these hotter or cooler than the Sun? Can you suggest a reason for the difference between this answer and the answer to the previous question? (Hint: Look at the luminosities.) Is there any tendency for a correlation between temperature and luminosity? Are there exceptions to the correlation?

What star appears the brightest in the sky (other than the Sun)? The second brightest? What color is Betelgeuse? Use Appendix J to find the answers.

Suppose hominids one million years ago had left behind maps of the night sky. Would these maps represent accurately the sky that we see today? Why or why not?

Why can only a lower limit to the rate of stellar rotation be determined from line broadening rather than the actual rotation rate? (Refer to [link].)

Why do you think astronomers have suggested three different spectral types (L, T, and Y) for the brown dwarfs instead of M? Why was one not enough?

Sam, a college student, just bought a new car. Sam’s friend Adam, a graduate student in astronomy, asks Sam for a ride. In the car, Adam remarks that the colors on the temperature control are wrong. Why did he say that?

(credit: modification of work by Michael Sheehan)

Would a red star have a smaller or larger magnitude in a red filter than in a blue filter?

Two stars have proper motions of one arcsecond per year. Star A is 20 light-years from Earth, and Star B is 10 light-years away from Earth. Which one has the faster velocity in space?

Suppose there are three stars in space, each moving at 100 km/s. Star A is moving across (i.e., perpendicular to) our line of sight, Star B is moving directly away from Earth, and Star C is moving away from Earth, but at a 30° angle to the line of sight. From which star will you observe the greatest Doppler shift? From which star will you observe the smallest Doppler shift?

What would you say to a friend who made this statement, “The visible-light spectrum of the Sun shows weak hydrogen lines and strong calcium lines. The Sun must therefore contain more calcium than hydrogen.”?

### Figuring for yourself

In Appendix J, how much more luminous is the most luminous of the stars than the least luminous?

You have enough information from this chapter to estimate the distance to Alpha Centauri, the second nearest star, which has an apparent magnitude of 0. Since it is a G2 star, like the Sun, assume it has the same luminosity as the Sun and the difference in magnitudes is a result only of the difference in distance. Estimate how far away Alpha Centauri is. Describe the necessary steps in words and then do the calculation. If you assume the distance to the Sun is in AU, your answer will come out in AU.

Do the previous problem again, this time using the information that the Sun is 150,000,000 km away. You will get a very large number of km as your answer. To get a better feeling for how the distances compare, try calculating the time it takes light at a speed of 299,338 km/s to travel from the Sun to Earth and from Alpha Centauri to Earth. For Alpha Centauri, figure out how long the trip will take in years as well as in seconds.

Star A and Star B have different apparent brightnesses but identical luminosities. If Star A is 20 light-years away from Earth and Star B is 40 light-years away from Earth, which star appears brighter and by what factor?

Star A and Star B have different apparent brightnesses but identical luminosities. Star A is 10 light-years away from Earth and appears 36 times brighter than Star B. How far away is Star B?

Our Sun, a type G star, has a surface temperature of 5800 K. We know, therefore, that it is cooler than a type O star and hotter than a type M star. Given what you learned about the temperature ranges of these types of stars, how many times hotter than our Sun is the hottest type O star? How many times cooler than our Sun is the coolest type M star?

## Facts

Deneb belongs to the spectral class A2 Ia. It is a blue-white supergiant, and one of the most luminous stars known. Among the 30 brightest stars in the sky, Deneb is by far the most distant, by a factor of almost 2. The star’s estimated diameter is 100 to 200 times that of the Sun, which makes it one of the largest A-type stars known.

Deneb is the prototype of a class of stars known as Alpha Cygni variables. These stars exhibit non-radial fluctuations of the surface and as a result their spectral type and luminosity change slightly.

Image showing the approximate size of the Sun relative to the much larger Deneb. Estimates for Deneb’s radius range from 200 to 300 times that of the Sun. This image splits the difference and shows Deneb with a radius 250 times that of the Sun.

The star has a mass about 19 times that of the Sun. It will have a relatively short life and probably go out in a supernova explosion within the next few million years. In its current phase, Deneb is likely expanding into a red supergiant. It is losing mass at a rate of 0.8 millionth of a solar mass every year as a result of a strong stellar wind.

Cygnus constellation represents the Swan. In Greek mythology, the constellation represents Zeus, who turned himself into a swan to seduce Queen Leda of Sparta. In another version of the tale, Zeus turned himself into a swan to trick the goddess Nemesis into giving him shelter. The two had an affair and Nemesis produced an egg which was then given to Leda by Hermes. It was said that Leda’s daughter Helen hatched out of the egg. She later became famous as Helen of Troy.

In the better known version of the myth, Zeus seduced Leda, who was married to King Tyndareus, and she gave birth to two sets of twins: Helen and Clytemnestra, and Castor and Polydeuces. Helen and Polydeuces were the immortal children of Zeus, and Clytemnestra and Castor were Tyndareus’ children and therefore mortal. The twins Castor and Polydeuces are represented by Gemini constellation, where the two brightest stars are named after them. Polydeuces is usually better known by his Latin name, Pollux.

Deneb – Alpha Cygni
Constellation: Cygnus
Location: 20h 41m 25.9s (right ascension), +45°16󈧵” (declination)
Spectral class: A2 Ia
Visual magnitude: 1.25
Absolute magnitude: -8.38
Mass: 19 ± 4 solar masses
Luminosity: 196,000 ± 32,000 solar
Temperature: 8,525 ± 75 K
Distance: 3,550 light years (802 parsecs) .
Variable type: Alpha Cygni
Designations: Deneb, α Cygni, 50 Cygni, Arided, Aridif, Gallina, Arrioph, HR 7924, BD +44°3541, HD197345, SAO 49941, FK5: 777, HIP 102098

## Travel Time

The time it will take to travel to this star is dependent on how fast you are going. U.G. has done some calculations as to how long it will take going at differing speeds. A note about the calculations, when I'm talking about years, I'm talking non-leap years only (365 days).

The New Horizons space probe is the fastest probe that we've sent into space at the time of writing. Its primary mission was to visit Pluto which at the time of launch (2006), Pluto was still a planet.

Mach 1 is the speed of sound, Mach 2 is twice the speed of sound. Corncorde before it was retired was the fastest commercial airline across the Atlantic and only one that could do Mach 2.

 Description Speed (m.p.h.) Time (years) Walking 4 57,865,832,374.84 Car 120 1,928,861,079.16 Airbus A380 736 314,488,219.43 Mach 1 767.269 301,671,681.64 Mach 2 1,534.54 150,835,644.23 New Horizons 33,000 7,014,040.29 Speed of Light 670,616,629.00 345.15

## Hercules's 5 Brightest Stars

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## Distances from Spectral Types

As satisfying and productive as variable stars have been for distance measurement, these stars are rare and are not found near all the objects to which we wish to measure distances. Suppose, for example, we need the distance to a star that is not varying, or to a group of stars, none of which is a variable. In this case, it turns out the H–R diagram can come to our rescue.

If we can observe the spectrum of a star, we can estimate its distance from our understanding of the H–R diagram. As discussed in Analyzing Starlight, a detailed examination of a stellar spectrum allows astronomers to classify the star into one of the spectral types indicating surface temperature. (The types are O, B, A, F, G, K, M, L, T, and Y each of these can be divided into numbered subgroups.) In general, however, the spectral type alone is not enough to allow us to estimate luminosity. A G2 star could be a main-sequence star with a luminosity of 1 LSun, or it could be a giant with a luminosity of 100 LSun, or even a supergiant with a still higher luminosity.

We can learn more from a star’s spectrum, however, than just its temperature. Remember, for example, that we can detect pressure differences in stars from the details of the spectrum. This knowledge is very useful because giant stars are larger (and have lower pressures) than main-sequence stars, and supergiants are still larger than giants. If we look in detail at the spectrum of a star, we can determine whether it is a main-sequence star, a giant, or a supergiant.

Suppose, to start with the simplest example, that the spectrum, color, and other properties of a distant G2 star match those of the Sun exactly. It is then reasonable to conclude that this distant star is likely to be a main-sequence star just like the Sun and to have the same luminosity as the Sun. But if there are subtle differences between the solar spectrum and the spectrum of the distant star, then the distant star may be a giant or even a supergiant.

The most widely used system of star classification divides stars of a given spectral class into six categories called luminosity classes. These luminosity classes are denoted by Roman numbers as follows:

• Ia: Brightest supergiants
• Ib: Less luminous supergiants
• II: Bright giants
• III: Giants
• IV: Subgiants (intermediate between giants and main-sequence stars)
• V: Main-sequence stars

The full spectral specification of a star includes its luminosity class. For example, a main-sequence star with spectral class F3 is written as F3 V. The specification for an M2 giant is M2 III. Figure 1 illustrates the approximate position of stars of various luminosity classes on the H–R diagram. The dashed portions of the lines represent regions with very few or no stars.

Figure 1: Luminosity Classes. Stars of the same temperature (or spectral class) can fall into different luminosity classes on the Hertzsprung-Russell diagram. By studying details of the spectrum for each star, astronomers can determine which luminosity class they fall in (whether they are main-sequence stars, giant stars, or supergiant stars).

With both its spectral and luminosity classes known, a star’s position on the H–R diagram is uniquely determined. Since the diagram plots luminosity versus temperature, this means we can now read off the star’s luminosity (once its spectrum has helped us place it on the diagram). As before, if we know how luminous the star really is and see how dim it looks, the difference allows us to calculate its distance. (For historical reasons, astronomers sometimes call this method of distance determination spectroscopic parallax, even though the method has nothing to do with parallax.)

The H–R diagram method allows astronomers to estimate distances to nearby stars, as well as some of the most distant stars in our Galaxy, but it is anchored by measurements of parallax. The distances measured using parallax are the gold standard for distances: they rely on no assumptions, only geometry. Once astronomers take a spectrum of a nearby star for which we also know the parallax, we know the luminosity that corresponds to that spectral type. Nearby stars thus serve as benchmarks for more distant stars because we can assume that two stars with identical spectra have the same intrinsic luminosity.

## Selected Eridanus's Random Stars

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