1. Stellar distances

2. Stellar Parallax

4. Parallax angle (Ɵ)
1 a.u.

6. Parallax Method January very distant reference star Sun ‘near’ star
Sun
‘near’ star
very distant
reference star
June

7. Very small angles

8. Parallax and Distance
The distance to a star is inversely related to its parallax, i.e. nearer stars have larger parallaxes. The relation between parallax  and distance d is:
d = 1/
Where, if  is measured in arcseconds, d is then determined in parsecs.
Thus, a star whose parallax is 1.0 arcseconds has a distance of 1.0 parsecs; one with  = 0.01 seconds of arc has a distance of 1/0.01 = 100 parsecs, etc.
One parsec equals 3.26 light years (the distance light travels in one year) or about 2x1013 miles! That is more than 200,000 AU.

9. Parsec
If the parallax angle is one arcsecond (1/3600 degree in radians) the distance to the star is called a parsec (pc)

10. Problems
1) The star Proxima Centauri has a parallax of arcsecs (0.762”). Calculate the distance to the star in i) parsecs (pc), ii) light years (ly) 2) Friedrich Bessel is credited as the first person to measure the distance to a star using trigonometric parallax. He measured the value of p=8.06 x 10-5 degrees for the star 61 Cygnus. Calculate the distance to 61 Cygnus in i) parsecs (pc), ii) light years (ly) 1 parsec = 3.26 ly

11. Some of the Brightest Stars
The Five Nearest Stars
Star Parallax Distance
arcsec pc
a Centauri
Barnard’s Star
Wolf
Lalande
Sirius
Some of the Brightest Stars
Star Parallax Distance
arcsec pc
Sirius
Canopus
a Centauri
Arcturus
Vega
Capella
Betelgeuse
Deneb
Note that the brightest stars are not necessarily the nearest. What does that mean?

12. Apparent and absolute magnitudes

13. Hipparchus
Greek astronomer
Lived 2000 years ago

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

15. Apparent magnitude and brightness
Magnitude 1 star is
100 times brighter than
a magnitude 6 star

16. 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
Magnitude 1
Magnitude 2
Magnitude 3
Magnitude 4
r5 = 100
Magnitude 5
r = 2.512
Magnitude 6

17. * Under what conditions?
Clear sky
When viewed from earth
As visible to the naked eye

18. Can a star have a magnitude greater than 6?

19. Can a star have a magnitude greater than 6?
Yes, but these stars are only visible through a telescope

20. Negative apparent magnitude?
They are very bright!!

21. Guess the apparent magnitude of Sun
It is -26.7

22. This is Pogson’s Equation
Apparent magnitude
The apparent magnitude m, of a star of apparent brightness I is defined by
m = -2.5log10I+K
where K is set by assigning a magnitude to a specific reference star.
This is Pogson’s Equation

23. Differences in intensity
One type of question often encountered is to compare the brightness of two stars, given their apparent magnitudes

24. Example
Apparent magnitude of Sun is and that of Betelgeuse is 0.5. How much brighter is Sun than Betelgeuse?

25. 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 = 27.2
Each difference in magnitude is a difference of in brightness
Therefore the difference in brightness =
= 7.6 x 1010

26. Sun is 76 billion times brighter than Betelgeuse

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

28. Apparent Magnitude
Apparent magnitude is how bright an object appears to us
This will be affected by 1) how bright the objects actually is (absolute magnitude) and 2) how far away the object is.
E.g. a dim object close to us could have the same apparent magnitude as a bright object far away.

29. Absolute magnitude
is the apparent
magnitude of a star
when viewed from
a distance of 10 parsecs.

30. Absolute magnitude

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

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

33. 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

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

35. Can absolute magnitude be
Positive ?
Negative ?
Any value?

36. Let’s try some questions!
‘Apparent and absolute magnitude questions’

37. Spectroscopic parallax

38. Spectroscopic parallax
This refers to the method of finding the distance to a star given the star’s luminosity (total power output of star in W) and apparent brightness (intensity as seen at the Earth in Wm-2). It doesn’t use parallax! Limited to distances less than 10 Mpc
We know that b = L/(4πd2) so d = (L/(4 πb))½

39. 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?
λ 0T = 2.9 x 10-3 Km
T = 2.9 x 10-3 / 4.3 x = 12000K

40. 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)

41. 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

42. Question
The peak wavelength of a main sequence star is 720nm. Its apparent brightness is measured as 2 x 10-9Wm-2.
What is:
a) its surface temperature?
b) its luminosity?
c) its distance from the Earth?

43. The same technique can be used for magnitudes
The light from a main sequence star has a peak wavelength of 100nm, and an apparent magnitude of 5
What is:
a) its surface temperature?
b) its distance from the Earth in parsecs?
c) its radius?

44. Using cepheids to measure distance

45. 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.

46. 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.

47. 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.

48. Cepheid calculation - Example

49. 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).

50. 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

51. 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