1. Stellar distances

2. Parallax

4. Parallax

5. Parallax angle

6. Parallax angle P (in rads) R (=1 AU) d tan p = R/d
for small p, tan p ≈ p
so d = R/p

7. Parsec
If the parallax angle is one arcsecond (1 “) the distance to the star is called a parsec

8. Parsec
If the parallax angle is one arcsecond (1 “) the distance to the star is called a parsec
d (parsecs) =
p (in arcsecs)

9. Parsec
1 pc = 3.26 ly

10. Example
A star has a parallax angle of 0.34 arcsecs. How far is the star away from earth in light years?
Distance in light years = 2.9 pc x 3.26 ly/pc = 9.5 ly

11. Converting degrees to arcsecs in radians
Multiply by 2π to convert to radians
360
Multiply by to convert to arcsecs
3600

12. Parallax method
Only useful for close stars (up to 300 ly ( or 100 pc) as further than that the parallax angle is too small (space based telescopes can use this method to measure stars up to distances of 500 pc). so another technique was required.

13. Apparent and Absolute magnitudes

14. Hipparchus Greek astronomer
Lived 2000 years ago, believed in the Geocentric system.

15. Hipparchus compared the relative brightness of stars (as seen from earth)
Brightest stars – magnitude 1
Second brightest – magnitude 2
...
Faintest star – magnitude 6

16. The Magnitude System
Apparent magnitude describes the relative brightness of objects as they appears in sky.
A difference of 5 magnitudes is equivalent to a factor of 100 difference in apparent brightness.
 1st magnitude star is 100 times brighter than a 6th magnitude star.
A difference of one magnitude is a factor of 2.51 difference in brightness.
The larger the magnitude, the fainter the object
Objects with negative magnitude appear brighter than objects with positive apparent magnitude.
Apparent magnitude mv of selected objects :
The brightest star in the in night time sky, Sirius, is mv = -1.4
The Sun: mv = -27
The full Moon is -13
Maximum brightness of
Venus: mv = -4.7
Mars: mv = -2.9
Jupiter: mv = -2.8
Large Magellantic Cloud: mv = 0.9
Andromeda galaxy: mv = 4.3
Faintest star visible to human eyes: mv = 6

17. The Absolute Magnitude
A star’s absolute magnitude Mv is the apparent magnitude it would have if it were at a distance of 10 parsecs (32.6 light-years) from Earth.
The Sun’s absolute magnitude is Mv = 4.8
Sirius: Mv = +1.4
Betelgeuse: Mv = -5.1
Apparent magnitude tells us nothing about the luminosity of the objects, but it tell us how difficult it is to see the objects in the sky.
Absolute magnitude, on the other hand, is directly related to the luminosity of the object. But it does not tell us how bright they appear in the sky.

18. The difference between a magnitude 1 star and a magnitude 6 star is ‘5 steps’ on the magnitude scale and the scale is logarithmic. This means that each ‘step’ equated to a brightness decrease of since
(2.512)5=100

19. Negative apparent magnitude?
Objects can have negative magnitudes
Negative apparent magnitude?
They are very bright!!

20. Apparent magnitude m = -(5/2)log (b/b0)
The apparent magnitude m, of a star of apparent brightness b is defined by
m = -(5/2)log (b/b0)
where b0 is taken as a reference value of 2.52 x 10-8 W.m-2
This can also be written as b/b0 = m

21. Question
Apparent magnitude of Sun is and that of Betelgeuse is 0.5. How much brighter is Sun than Betelgeuse?

22. Difference in magnitudes is 0.5 – (-26.7) = 27.2
Apparent magnitude of Sun is and that of Betelgeuse is 0.5. How much brighter is Sun than Betelgeuse?
Difference in magnitudes is 0.5 – (-26.7) = 27.2
Each difference in magnitude is a difference of in brightness ((2.512)5=100 )
Therefore the difference in brightness
= = 7.6 x 1010 times brighter.

23. Sun is 76 billion times brighter than Betelgeuse

24. Question 2
Apparent magnitudes of Andromeda galaxy and Crab nebula are 4.8 and 8.4 respectively.
Which of these is brightest?
By what factor?

25. Difference in apparent magnitudes = 8.4 – 4.8 = 3.6
Galaxy is brighter
Difference in apparent magnitudes = 8.4 – 4.8 = 3.6
Difference in brightness therefore = = 27.5 times

26. The Andomeda Galaxy is a vast collection
of stars
The Crab Nebulae is a debris
of supernova and is the
birth place of the new star.

27. Apparent magnitude and distances
Is it a fair way of measuring brightness of a star?
Brightness depends on distance and obeys inverse square law

28. ABSOLUTE MAGNITUDE Let the standard distance be 10 pc
1 pc = x 1016 m
= 3.26 ly
= AU

29. Recall: Absolute magnitude
is the apparent
magnitude of a star
when viewed from
a distance of 10 pc.

30. Absolute magnitude M and apparent magnitude m
m – M = 5 log (d/10)
d is in parsecs!

31. Question Calculate the absolute magnitude of Sun.
Apparent magnitude = -26.7
Distance from earth = 4.9 x 10-6 pc

32. m – M = 5 log(d/10)
M = 5 log (4.9 x 10-6/10)
M =-26.7 – 5log(4.9 x 10-7)
M = 4.85

33. M = 4.85
This means at a standard distance of 10 parsecs the sun would appear to be a dim star.

34. Recall Determination of Distance
Stellar Parallax
Knowledge of the distance to the stars is crucial for our determination of the luminosity of stars…
Current technology allows us to determine the distance accurately to within a few hundred light-years.
Hipparcos mission (European Space Agency) measured the stellar parallax of roughly 100,000 stars with precision of a few milli-arcseconds. So, it can measure distance of star up to 1,000 light-years away…
Simulation of Stellar Parallax…

35. Luminosity
To directly measure the luminosity of a star (let’s say, the Sun), we will need to surround the Sun completely with detectors, which is impossible.
We can infer the luminosity of the Sun if we know
the distance to the star, and
the star’s apparent brightness
Further more, we need to assume that
the star emits energy uniformly in all direction…
Then we can calculate its luminosity by the formula: d
The total area of the sphere with a radius of r is 4d2

36. Quiz: Which Star Has Higher Luminosity?
Apparent Brightness B
Distance d A 10 1 B
The apparent brightness decrease as d 2
The brightness of star A is 10 × 1 = 10
The brightness of star B is 1 × 102 = 100 if observed at distance 1
 Star B is 10 times more luminous than A!
The photons contained in box A are spread into an area 4 times as large in box Y which is twice the distance from the star as X. X Y

37. Luminosity of Selected Stars
Distance
[ly]
Spectral
Type
Luminosity
[L/Lsun]
Prosima Centauri
4.2
M5.5
0.0006
Bernard’s Star
6.0 M4
0.005
Gliese 725 A
11.4 M3
0.02
 Centauri B
4.4 K0
0.53
Sun G2
1.0
 Centauri A
1.6
Sirius A
8.6 A1
26.0
Vega 25 A0 60
Achernar
144 B5
3,600
Betelgeuse
423 M2
38,000
Deneb
2500 A2
170,000

38. Luminosity and Distance — Chicken and Egg
Most of the time, we need measurement of distance to calculate the luminosity. Howver, if we can determine the luminosity of an object with other methods (independent of distance measurement, such as the luminosity of supernovae), then we can derive the distance to the object from measurement of their apparent brightness.

39. Astronomical Distance Units
Light-year:
The distance light travels (in vacuum) in one year.
one light-year is 10 trillion (1013) km
Parsec: parallax & arcsecond
One parsec: the distance to an object with a parallax angle of 1 arcsecond.
One parsec equals to 3.26 light-year.
kiloparsecs: 1,000 parsecs.
megaparsecs: 1,000,000 parsec.

40. Spectroscopic parallax

41. Spectroscopic parallax
This refers to the method of finding the distance to a star given the star’s luminosity and apparent brightness. It doesn’t use parallax! Limited to distances less than 10 Mpc
We know that b = L/(4πd2) so d = (L/(4 πb))½

42. Spectroscopic parallax - Example
A main sequence star emits most of its energy at λ = 2.4 x 10-7 m. Its apparent brightness is measured to be 4.3 x 10-9 W.m-2. How far away is the star?
Wein’s Law: λ 0T = 2.9 x 10-3 Km
so, T = 2.9 x 10-3 / 2.4 x = 12000K
Wein’s Law to get temperature

43. T = 12000K. From an HR diagram we can see this corresponds to a brightness of about 100x that of the sun (= 100 x 3.9 x 1026 = 3.9 x 1028 W)
The area is A = 4πR2
A = 4(3.1416)(7 x 108)2
A = 6.16 x 1018 m2
The flux is W = σ T4
W = × 10-8 (5780)4
W = 6.33 x 107 Watts/m2
The total power is P = WA= (5.19 x 107 Watts/m2)(6.16 x 1018 m2 )
P = 3.9 x 1026 Watts

44. Spectroscopic parallax - Example
Thus d = (L/(4 πb))½
d = (3.9 x 1028/(4 x π x 4.3 x 10-9))½
d = 8.5 x 1017 m = 90 ly = 28 pc

45. Hertzsprung-Russell Diagram
Sizes scale
1 Rsun
10 Rsun
100 Rsun
1000 Rsun
Since there appears to be a strong correlation between luminosity and color (temperature), we put all the stars on a Luminosity – Temperature plot, and this is what it looks like:
Properties of Stars shown in the H-R Diagram:
Luminosity (log scale).
Temperature and spectral type
Size
Mass of the main sequence
Lifetime

46. Hertzsprung-Russell Diagram
Sizes scale
1 Rsun
10 Rsun
100 Rsun
1000 Rsun
Notice that…
Temperature scale decreases from left to right.
The scale of luminosity is in power of 10 (log scale).
Mass increases from lower right to upper left
Size increases from lower left to upper right.

47. Classification of Stars in H-R Diagram
Sizes scale
1 Rsun
10 Rsun
100 Rsun
1000 Rsun
The Main Sequence stars
healthy stars, fusing hydrogen in the core.
High-mass, high-luminosity, high-temperature, and short-lived stars on the upper-left-hand corner
Low-mass, low-luminosity, low-temperature, and long-lived stars on the lower-right-hand corner
The Supergiants,
The Giants,
Supergiants and giants are dying stars, fusing helium and heavier elements.
The White Dwarfs.
dead stars, exposed core of dead main-sequence stars.

48. Cepheid variables
At distances greater than Mpc, neither parallax nor spectroscopic parallax can be relied upon to measure the distance to a star.
When we observe another galaxy, all of the stars in that galaxy are approximately the same distance away from the earth. What we really need is a light source of known luminosity in the galaxy. If we had this then we could make comparisons with the other stars and judge their luminosities. In other words we need a ‘standard candle’ –that is a star of known luminosity.
The outer layers of Cepheid variable stars undergo periodic expansion and contraction, producing a periodic variation in its luminosity.

49. Cepheid variable stars are useful to astronomers because of the period of their variation in luminosity turns out to be related to the average absolute magnitude of the Cepheid. Thus the luminosity of the Cepheid can be calculated by observing the variation in brightness.

50. The process of estimating the distance to a galaxy (in which the individual stars can be imagined) might be as follows:
Locate a Cepheid variable in the galaxy
Measure the variation in brightness over a given period of time.
Use the luminosity-period relationship for Cepheids to estimate the average luminosity.
Use the average luminosity, the average brightness and the inverse square law to estimate the distance to the star.

51. Cepheid calculation - Example

52. From the left-hand graph we can see that the period of the cepheid is 5.4 days. From the second graph we can see that this corresponds to a luminosity of about 103 suns (3.9 x 1029 W).

53. From the left hand graph we can see the peak apparent magnitude is 3
From the left hand graph we can see the peak apparent magnitude is 3.6 which means we can find the apparent brightness from
b/b0 = m
b = 2.52 x 10-8 x = 9.15 x W.m-2

54. Now using the relationship between apparent brightness, luminosity and distance
d = (L/(4πb))½
d = (3.9 x 1029/(4 x π x 9.15 x 10-10))½
d = 5.8 x 1018 m = 615 ly = 189 pc

55. Determination of Stellar Mass
Mass is the single most important property of a star. But it is also difficult to measure…
The most dependable method we have for measuring the mass of distant stars is Newton’s version of Kepler’s Third Law of orbital motion.
Recall that
So, if we can find
two stars (binary star system) orbiting each other, and
if we can measure their
rotational period p, and
semi-axis a of the orbit,
then we can determine their masses.

56. Binary Star Systems
Binary star systems are formed by two stars that are gravitationally bounded, and they orbit each other.
About 50% of the stars are in binary star system. There are three categories of binary star systems:
Visual Binary: a pair of stars that we can see distinctly (with a telescope) as the stars orbit each other.
Eclipsing Binary: is a pair of stars that orbit in the plane of our line of sight. The stars are not resolved, but we can see the effects of the stars blocking each other in their combined light-curve.
Spectroscopic Binary: in some binary system, we cannot see the two stars, nor can we see their light curve changes, but we can see the motion of the stars from Doppler effect measurement of the spectra. B
Center of mass A
True Binary Star System

57. Binary Star Systems
Two stars appearing close to each other in the sky do not necessarily means that they are a binary system. B
Line-of-Sight A
If A and B are not
gravitationally bounded with each other, then, although they may appears to be very close in the sky, they do not constitute a binary system! A B

58. Visual Binary – Sirius
Sirius (in constellation Canis Major) is the brightest star in the night-time sky (magnitude -1.4). It is a visual binary system. Sirius A (the larger of the two) is a main sequence star with spectral type A0, and Sirius B is a white dwarf.
Hubble Space Telescope image of
Sirius
Sirius A & B time sequence

59. Eclipsing Binary
About 50% of the stars are in binary star system. There are three categories of binary star systems:
Eclipsing Binary: is a pair of stars that orbit in the plane of our line of sight, (measuring the time curve)

60. Algol – Eclipsing Binary
Algol (the demon star) is in the constellation of Perseus.
Algol A: main sequence star, more massive.
Algol B: subgiant, less massive.
The Algol Paradox:
Why is the more massive Algol A evolve slower than the less massive Algol B
Flow of material from A to B

61. Spectroscopic Binary
Sometimes only the spectrum from one star is seen, the other star is too dim.
Sometimes two sets of spectra can be seen at the same time
Sometimes more than two sets of spectra can be seen
Mizar is a visual binary system in the constellation of Big Dipper.
Each ‘star’ in the visual binary system is also a spectroscopic binary!

62. Eclipsing Binary and Stellar Mass Measurements
Among the three types of binary star systems, the eclipsing binary system is most important for the determination of stellar mass, because
Determination of the stellar mass requires knowledge of the orbital period and distance (in real distance unit, not in angular separation).
Orbital period is easy to measure, but distance between the stars is difficult to determine.
For visual binary, we need to know the distance from Earth to the stars before we can determine the separation between the stars in the binary system.
For spectroscopic binary, we can calculate the separation between the stars if we know their orbital speed. However, we can only determine the line-of-sight speed of the binary system from Doppler measurement. If the orbits are tilted with respect to our line-of-sight, then we under estimate the orbital speed.
If an eclipsing binary is also a spectroscopic binary, then we know its true orbital speed, and can determine the separation between the two stars. Then, the masses of the stars can be determined!