1. Measuring the Properties of Stars
Arny, 3rd Edition, Chapter 12

2. Introduction
Those tiny glints of light in the night sky are in reality huge, dazzling balls of gas, many of which are vastly larger and brighter than the Sun
They look dim because of their vast distances
Astronomers cannot probe stars directly, and consequently must devise indirect methods to ascertain their intrinsic properties
Measuring the Properties of Stars

3. Measuring a Star’s Distance
Introduction
Measuring distances to stars and galaxies is not easy
Distance is very important for determining the intrinsic properties of astronomical objects
Measuring Distance by Triangulation and Parallax
Fundamental method for measuring distances to nearby stars is triangulation:
Measure length of a triangle’s “baseline” and the angles from the ends of this baseline to a distant object
Use trigonometry or a scaled drawing to determine distance to object
A method of triangulation used by astronomers is called parallax:
Baseline is the Earth’s orbit diameter (2 AU)
Angles measures with respect to very distant stars
Figure 12.1, Figure 12.2
Animation: Parallax is used to measure the distance to stars
Measuring the Properties of Stars

4. Measuring a Star’s Distance
Measuring Distance by Triangulation and Parallax (continued)
The shift in position of nearby stars is very small – angles are measured in arc seconds
Defining the parallax angle, p, as half the angular shift of the nearby star, its distance in parsecs is given by:
dpc = 1/parc seconds
A parsec is 3.26 light-years (3.09x1013 km)
Due to the very small parallax angles of distant stars, the parallax method is useful only to distances of about 250 parsecs
Measuring the Distance to Sirius
Measured parallax angle for Sirius is arc second
From formula dpc = 1/0.377 = 2.65 parsecs = 8.6 light-years
Box Figure 12.1
Measuring the Properties of Stars

5. Measuring a Star’s Distance
Measuring Distance by the Standard-Candles Method
If an object’s intrinsic brightness is known, its distance can be determined from its observed brightness
Astronomers call this method of distance determination the method of standard candles
The “trick” to the method is to know in advance what the intrinsic brightness of the object is and then use the inverse square law to find the distance
This method is the principle manner in which astronomers determine distances in the universe
The various “standard candles” will be discussed as appropriate
Measuring the Properties of Stars

6. Measuring the Properties of Stars from Their Light
Introduction
Astronomers want to know the motions, sizes, colors, and structures of stars
This information helps understand the nature of stars as well as their life cycle
The light from stars received at Earth is all that is available for this analysis
Temperature
The color of a star indicates its relative temperature – blue stars are hotter than red stars
More precisely, a star’s surface temperature is given by Wien’s law
T = 3x106/lm
where T is the star’s surface temperature in Kelvin and lm is the wavelength in nanometers (nm) at which the star radiates most strongly
Figure 12.3
Measuring the Properties of Stars

7. Measuring the Properties of Stars from Their Light
Luminosity
The amount of energy a star emits each second is its luminosity (usually abbreviated as L)
A typical unit of measurement for luminosity is the watt
Compare a 100-watt bulb to the Sun’s luminosity, 4x1026 watts
Luminosity is a measure of a star’s energy production (or hydrogen fuel consumption)
Knowing a star’s luminosity will allow a determination of a star’s distance and radius
Measuring the Properties of Stars

8. Measuring the Properties of Stars from Their Light
The Inverse - Square Law and Measuring a Star’s Luminosity
The inverse–square law relates an object’s luminosity to its distance and its apparent brightness (how bright it appears to us)
The inverse-square law can physically be thought of as the result of a fixed number of photons, spreading out evenly in all directions as they leave their source, having to cross larger and larger spherical shells, each centered on the source – for a given shell, the number of photons decreases per unit area
The inverse-square law is:
B = L/(4pd2)
where B is the brightness at a distance d from a source of luminosity L
Figure 12.4
Animation: The inverse-square law
Measuring the Properties of Stars

9. Measuring the Properties of Stars from Their Light
The Inverse - Square Law and Measuring a Star’s Luminosity (continued)
This relationship is called the inverse-square law because the distance appears in the denominator as a square
The inverse-square law is one of the most important mathematical tools available to astronomers:
Given d from parallax measurements, a star’s L can be found (A star’s B can easily be measured by an electronic devise, called a photometer, connected to a telescope)
Or if L is known in advance, a star’s distance can be found
Measuring the Properties of Stars

10. Measuring the Properties of Stars from Their Light
Radius
Common sense: Two objects of the same temperature but different sizes, the larger one radiates more energy than the smaller one
In stellar terms: a star of larger radius will have a higher luminosity than a smaller star at the same temperature
The Stefan – Boltzmann Law
To put our common sense feeling of temperature, luminosity, and radius into an equation, we first need to know how much energy is emitted per unit area of a surface held at a certain temperature
The Stefan-Boltzmann law gives this: sT4, where s is the Stefan-Boltzmann constant (5.67x10-8 watts m-2K-4)
Measuring the Properties of Stars

11. Measuring the Properties of Stars from Their Light
The Stefan – Boltzmann Law (continued)
The Stefan-Boltzmann law only applies to “objects of substance” – it applies to stars, but not hot, low-density gases
Now, our common sense feeling of temperature, luminosity, and radius can be expressed as an equation :
L = 4pR2sT4
where R is the radius of the star (measured in meters when L is in watts, T in Kelvin, and s in watts m-2K-4)
Given L and T, we can then find a star’s radius
For several dozen nearby stars and a few very large stars, their radii can be determined more directly by interferometers
Figure 12.5, Figure 12.6
Measuring the Properties of Stars

12. Measuring the Properties of Stars from Their Light
The Stefan – Boltzmann Law (continued)
The methods using the Stefan-Boltzmann law and interferometer observations show that stars differ enormously in radius
Some stars are hundreds of times larger than the Sun and are referred to as giants
Stars smaller than the giants are called dwarfs
Measuring the Radius of the Star Sirius
Solving for a star’s radius can be simplified if we apply L = 4pR2sT4 to both the star and the Sun, divide the two equations, and solve for radius:
Rs/R¤ = (Ls/L¤)1/2(T¤/Ts)2
Where s refers to the star and ¤ refers to the Sun
Given for Sirius Ls=25L¤, Ts= 10,000 K, and for the Sun T¤=6000 K, one finds Rs=1.8R¤
Measuring the Properties of Stars

13. Measuring the Properties of Stars from Their Light
The Magnitude System
About 150 B.C., the Greek astronomer Hipparchus measured apparent brightness of stars using units called magnitudes
Brightest stars had magnitude 1 and dimmest had magnitude 6
The system is still used today and units of measurement are called apparent magnitudes to emphasize how bright a star looks to an observer
A star’s apparent magnitude depends on the star’s luminosity and distance – a star may appear dim because it is very far away or it does not emit much energy
Measuring the Properties of Stars

14. Measuring the Properties of Stars from Their Light
The Magnitude System (continued)
The apparent magnitude can be confusing
Scale runs “backward”: high magnitude = low brightness
Modern calibrations of the scale create negative magnitudes
Magnitude differences equate to brightness ratios:
A difference of 5 magnitudes = a brightness ratio of 100
1 magnitude difference = brightness ratio of 1001/5=2.512
Astronomers use “absolute magnitude” to measure a star’s luminosity
The absolute magnitude of a star is the apparent magnitude that same star would have at 10 parsecs
A comparison of absolute magnitudes is now a comparison of luminosities, no distance dependence
An absolute magnitude of 0 approximately equates to a luminosity of 100L¤
Measuring the Properties of Stars

15. Spectra of Stars Introduction Measuring a Star’s Composition
A star’s spectrum typically depicts the energy it emits at each wavelength
A spectrum also can reveal a star’s composition, temperature, luminosity, velocity in space, rotation speed, and other properties
On certain occasions, it may reveal mass and radius
Measuring a Star’s Composition
As light moves through the gas of a star’s surface layers, atoms absorb radiation at some wavelengths, creating dark absorption lines in the star’s spectrum
Every atom creates its own unique set of absorption lines
Determining a star’s surface composition is then a matter of matching a star’s absorption lines to those known for atoms
Figure 12.7, Figure 12.8
Measuring the Properties of Stars

16. Spectra of Stars Measuring a Star’s Composition (continued)
To find the quantity of a given atom in the star, we use the darkness of the absorption line
This technique of determining composition and abundance is complicated by:
Possible overlap of absorption lines from several varieties of atoms being present
Temperature can also affect how strong (dark) an absorption line is
How Temperature Affects a Star’s Spectrum
A photon is absorbed when its energy matches the difference between two electron energy levels and an electron occupies the lower energy level
Higher temperatures, through collisions and energy exchange, will force electrons, on average, to occupy higher electron levels – lower temperatures, lower electron levels
Measuring the Properties of Stars

17. Spectra of Stars How Temperature Affects a Star’s Spectrum (continued)
Consequently, absorption lines will be present or absent depending on the presence or absence of an electron at the right energy level and this is very much dependent on temperature
Adjusting for temperature, a star’s composition can be found – interestingly, virtually all stars have compositions very similar to the Sun’s: 71% H, 27% He, and a 2% mix of the remaining elements
Classification of Stellar Spectra
Historically, stars were first classified into four groups according to their color (white, yellow, red, and deep red), which were subsequently subdivided into classes using the letters A through N
Annie Jump Cannon discovered the classes were more orderly in appearance if rearranged by temperature – Her reordered sequence became O, B, A, F, G, K, M (O being the hottest and M the coolest) and are today known as spectral classes
Cecilia Payne then demonstrated the physical connection between temperature and the resulting absorption lines
Measuring the Properties of Stars

18. Spectra of Stars Definition of Spectral Classes
A star’s spectral class is determined by the lines in its spectrum, but equally important, the lines tell us about the state of the atoms themselves
Some examples
O stars are very hot and the weak hydrogen absorption lines indicate that hydrogen is in a highly ionized state
A stars have just the right temperature to put electrons into hydrogen’s 2nd energy level, which results in strong absorption lines in the visible
F,G, and K stars are of a low enough temperature to show absorption lines of metals such as calcium and iron, elements that are typically ionized in hotter stars
K and M stars are cool enough to form molecules and their absorption “bands” become evident
Temperature range: more than 25,000 K for O (blue) stars and less than 3500 K for M (red) stars
Spectral classes subdivided with numbers - the Sun is G2
Figure 12.9
Measuring the Properties of Stars

19. Spectra of Stars Measuring a Star’s Motion
A star’s motion is determined from the Doppler shift of its spectral lines
The amount of shift depends on the star’s radial velocity, which is the star’s speed along the line of sight
Given that we measure Dl, the shift in wavelength of an absorption line of wavelength l, the radial speed v is given by:
v = (Dl /l)c
where c is the speed of light
Note that l is the wavelength of the absorption line for an object at rest and its value is determined from laboratory measurements on nonmoving sources
An increase in wavelength means the star is moving away, a decrease means it is approaching – speed across the line on site cannot be determined from Doppler shifts
Figure 12.10
Measuring the Properties of Stars

20. Spectra of Stars Measuring a Star’s Motion (continued)
Doppler measurements and related analysis show:
All stars are moving and that those near the Sun share approximately the same direction and speed of revolution (about 200 km/sec) around the center of our galaxy
Superposed on this orbital motion are small random motions of about 20 km/sec
In addition to their motion through space, stars spin on their axes and this spin can be measured using the Doppler shift technique – young stars are found to rotate faster than old stars
Figure 12.11
Measuring the Properties of Stars

21. Binary Stars Introduction
Two stars that revolve around each other as a result of their mutual gravitational attraction are called binary stars
Binary star systems offer one of the few ways to measure stellar masses – and stellar mass plays the leading role in a star’s evolution
At least 40% of all stars known have orbiting companions (some more than one)
Most binary stars are only a few AU apart – a few are even close enough to touch
Figure 12.12
Measuring the Properties of Stars

22. Binary Stars Visual and Spectroscopic Binaries
Visual binaries are binary systems where we can directly see the orbital motion of the stars about each other by comparing images made several years apart
Spectroscopic binaries are systems that are inferred to be binary by a comparison of the system’s spectra over time
Doppler analysis of the spectra can give a star’s speed and by observing a full cycle of the motion the orbital period can be determined
From the orbital period and speed, the size of the star’s orbit can be derived
Figure 12.13
Animation: Orbital motion is reflected in the shifting spectral lines of a spectroscopic binary
Measuring the Properties of Stars

23. Binary Stars Measuring Stellar Masses with Binary Stars
Kepler’s third law as modified by Newton is:
(m + M)P2 = a3
where m and M are the binary star masses (in solar masses), P is their period of revolution (in years), and a is the semimajor axis of one star’s orbit about the other (in AU)
P and a are determined from observations (may take a few years) and the above equation gives the combines mass (m + M)
Further observations of the stars’ orbit will allow the determination of each star’s individual mass
Most stars have masses that fall in the narrow range 0.1 to 30 M¤
Figure 12.14
Measuring the Properties of Stars

24. Binary Stars Eclipsing Binaries
A binary star system in which one star can eclipse the other star is called an eclipsing binary
Watching such a system over time will reveal a combined light output that will periodically dim
The duration and manner in which the combined light curve changes together with the stars’ orbital speed allows astronomers to determine the radii of the two eclipsing stars
Detailed investigation of the combined binary light curve also permits a determination of “star spots”
Figure 12.15
Measuring the Properties of Stars

25. Summary of Stellar Properties
Distance
Parallax (triangulation) for nearby stars (distances less than 250 pc)
Standard-candle method for more distant stars
Temperature
Wien’s law (color-temperature relation)
Spectral class (O hot; M cool)
Luminosity
Measure star’s apparent brightness and distance and then calculate with inverse square law
Luminosity class of spectrum (to be discussed)
Composition
Spectral lines observed in a star
Measuring the Properties of Stars

26. Summary of Stellar Properties
Radius
Stefan-Boltzmann law (measure L and T, solve for R)
Interferometer (gives angular size of star; from distance and angular size, calculate radius)
Eclipsing binary light curve (duration of eclipse phases)
Mass
Modified form of Kepler’s third law applied to binary stars
Radial Velocity
Doppler shift of spectrum lines
Measuring the Properties of Stars

27. The H-R Diagram Introduction
So far, only properties of stars have been discussed – this follows the historical development of studying stars
The next step is to understand why stars have these properties in the combinations observed
This step in our understanding comes from the H-R diagram, developed independently by Ejnar Hertzsprung and Henry Norris Russell in 1912
The H-R diagram is a plot of stellar temperature vs luminosity
Interestingly, most of the stars on the H-R diagram lie along a smooth diagonal running from hot, luminous stars (upper left part of diagram) to cool, dim ones (lower right part of diagram)
Figure 12.16
Measuring the Properties of Stars

28. The H-R Diagram Constructing the H-R Diagram
By tradition, bright star are placed at the top of the H-R diagram and dim ones at the bottom, while high-temperature (blue) stars are on the left with cool (red) stars on the right (Note: temperature does not run in a traditional direction)
The diagonally running group of stars on the H-R diagram is referred to as the main sequence
Generally, 90% of a group stars will be on the main sequence; however, a few stars will be cool but very luminous (upper right part of H-R diagram), while others will be hot and dim (lower left part of H-R diagram)
Figure 12.17
Measuring the Properties of Stars

29. The H-R Diagram Analyzing the H-R Diagram
The Stefan-Boltzmann law is a key to understanding the H-R diagram
For stars of a given temperature, the larger the radius is the larger the luminosity
Therefore, as one moves up the H-R diagram, a star’s radius must become bigger
On the other hand, for a given luminosity, the larger the radius is the smaller the temperature
Therefore, as one moves right on the H-R diagram, a star’s radius must increase
The net effect of this is that the smallest stars must be in the lower left corner of the diagram and the largest stars in the upper right
Stars in the upper left are called red giants (red because of the low temperatures there)
Stars in the lower right are white dwarfs
Three stellar types: main sequence, red giants, and white dwarfs
Figure 12.18
Measuring the Properties of Stars

30. where L and M are measured in solar units
The H-R Diagram
Giants and Dwarfs
Giants, dwarfs, and main sequence stars also differ in average density, not just diameter
Typical density of main-sequence star is 1 g/cm3, while for a giant it is 10-6 g/cm3
This difference is related to the large variation in volume for stars, but relatively low variation in stellar masses
The Mass-Luminosity Relation
Main-sequence stars obey a mass-luminosity relation, approximately given by:
L = M3
where L and M are measured in solar units
Consequence: Stars at top of main-sequence are more massive than stars lower down
Figure 12.19
Measuring the Properties of Stars

31. The H-R Diagram Luminosity Classes
Another method was discovered to measure the luminosity of a star (other than using a star’s apparent magnitude and the inverse square law)
It was noticed that some stars had very narrow absorption lines compared to other stars of the same temperature
It was also noticed that luminous stars had narrower lines than less luminous stars
Width of absorption line depends on density: wide for high density, narrow for low density
Luminous stars (in upper right of H-R diagram) tend to be less dense, hence narrow absorption lines
H-R diagram broken into luminosity classes: Ia (bright suoergiant), Ib (supergiants), II(bright giants), III (giants), IV(subgiants), V (main sequence)
Star classification example: The Sun is G2V
Figure 12.20, Figure 12.21
Measuring the Properties of Stars

32. The H-R Diagram Summary of the H-R Diagram
Most stars lie on the main sequence
Of these, the hottest stars are blue and more luminous, while the coolest stars are red and dim
Star’s position on sequence determines its mass, being more near the top of the sequence
Three classes of stars:
Main-sequence
Giants
White dwarfs
Measuring the Properties of Stars

33. Variable Stars Introduction
Not all stars have a constant luminosity – some change brightness: variable stars
There are several varieties of stars that vary and are important distance indicators
Especially important are the pulsating variables – stars with rhythmically swelling and shrinking radii
Radius change induces temperature and luminosity changes
For most pulsating variable stars, the temperature and luminosity variations are periodic – the repeating time interval for the variation is the variable star’s period
Variable stars are classified by the shape and period of their light curves – Mira and Cepheid variables are two examples
Most variable stars plotted on H-R diagram lie in the narrow “instability strip”
Figure 12.22, Figure 12.23
Measuring the Properties of Stars

34. Summary A Picture is worth a thousand words:
Figure 12.24
Measuring the Properties of Stars

35. Sketch illustrating the principle of triangulation.
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36. (a) Triangulation to measure a star's distance
(a) Triangulation to measure a star's distance. The radius of the Earth's orbit is the baseline. (b) As the Earth moves around the Sun, the star's position changes as seen against background stars. Parallax is defined as one half the angle by which the star's position shifts. Size of bodies and their separation are exaggerated for clarity.
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37. How to determine the relation between a star's distance and its parallax.
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38. Measuring a star's surface temperature using Wien's law.
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39. The inverse-square law
The inverse-square law. (a) Light spreads out from a point source in all directions. (b) As photons move out from a source, they are spread over a progressively larger area as the distance from the source increases. Thus a given area intercepts fewer photons the farther it is from the source. (c) The area over which light a distance d from the source is spread is 4pd2.
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40. The Stefan-Boltzmann law can be used to find a star's radius
The Stefan-Boltzmann law can be used to find a star's radius. (a) Each part of its surface radiates sT4. (b) Multiplying sT4 by the star's surface area (4pR2) shows that its total power output—its luminosity, L—is 4pR2sT4. To find the star's radius, we solve this equation to get R=[L/(4psT4)]1/2. Finally, we use this equation and measured values of the star's L and T to evaluate R.
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41. Image of the star Betelgeuse made by interferometry
Image of the star Betelgeuse made by interferometry. The lack of detail in this image reflects the current difficulty in observing the disk of any star but our Sun. One mas is approximately 2.8 x 10-7 degrees. (Courtesy Mullard Radio Obs).
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42. Computer generated spectra of three stars. A is hotter than the Sun
Computer generated spectra of three stars. A is hotter than the Sun. B is similar to the Sun. C is cooler than the Sun. (Courtesy Roger Bell and Michael Briley.)
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43. Formation of stellar absorption lines
Formation of stellar absorption lines. Atoms in the cooler atmospheric gas absorb radiation at wavelengths corresponding to jumps between electron orbits.
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44. The stellar spectral classes
The stellar spectral classes. (A computer generated depiction, courtesy Roger Bell and Michael Briley.)
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45. Measuring a star's radial velocity from its Doppler shift
Measuring a star's radial velocity from its Doppler shift. (a) Spectrum lines from a star moving away from Earth are shifted to longer wavelengths (a red shift). Spectrum lines from a star approaching Earth are shifted to shorter wavelengths (a blue shift). (b) A standard lamp attached inside the telescope serves as a comparison spectrum to allow the wavelength shift of the star's light to be found.
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46. Measuring a star's rotation velocity
Measuring a star's rotation velocity. As a star spins, one edge approaches, making a blue shift, and the other edge recedes, making a red shift. The result is that spectral lines are widened. The line's width can give the star's rotation velocity.
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47. Two stars orbiting in a binary system, held together as a pair by their mutual gravitational attraction.
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48. Spectroscopic binary star
Spectroscopic binary star. (a) The two stars are generally too close to be separated by even powerful telescopes. (b) Their orbital motion creates a different Doppler shift for the light from each star. Thus the spectrum of the stellar pair contains two sets of lines, one from each star.
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49. Measuring the combined mass of two stars in a binary system using the modified form of Kepler's third law.
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50. An eclipsing binary and its light curve
An eclipsing binary and its light curve. From the duration of the eclipse the stars' radii may be found. Note: In this illustration the blue star is hotter than the red star.
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51. The H-R diagram. The long diagonal line from upper left to lower right is the main sequence. Giant stars lie above the main sequence. White dwarf stars lie below it. Notice that bright stars are at the top of the diagram and dim stars at the bottom. Also notice that hot (blue) stars are on the left and cool (red) stars are on the right.
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52. Constructing an H-R diagram from the spectra and luminosities of stars in a cluster.
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53. Lines showing where in the H-R diagram stars of a given radius will lie.
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54. (a) The mass-luminosity relation shows that more massive stars are more luminous. (b) On the H-R diagram, high-mass stars lie higher on the main sequence than low-mass stars.
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55. Spectral lines are narrow in giant (luminous) stars (a and b) and wide in main-sequence (dim) stars (c). (Based on a figure courtesy Roger Bell and Michael Briley.)
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56. Stellar luminosity classes.
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57. Some variable star light curves
Some variable star light curves. (Courtesy AAVSO International Data Base and Douglas L. Welch, McMaster University).
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58. The instability strip in the H-R diagram
The instability strip in the H-R diagram. Stars that fall in this band generally pulsate.
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59. Summary of how astronomers measure the distance, temperature, mass, composition and radius of stars.
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