3. Measuring a distant cluster

4. Measuring the earth Going up!

5. Measuring the earth Still going up, but look far!

6. Measuring the earth Look up!

7. Measuring the earth

8. Measuring the solar system

9. Measuring the solar system Measuring angles

10. Measuring the solar system Continuing with Kepler

11. Measuring the solar system Measuring the curvature of the orbit of the moon! distance earth to moon distance sun to moon

12. But, but, but... Why the moon doesn't fly away? It is! Measuring the solar system

13. So what?

14. Measuring a distant cluster Trigonometric parallaxes

15. Measuring a distant cluster Spectroscopic parallaxes correlation calculation formula

16. Measuring a distant cluster Consider two stars of equal luminosity at distances D1 and D2, we can get ：

17. Measuring a distant cluster

18. NGC 188

19. Measuring a distant cluster (mass)

20. Measuring a distant cluster Turnoff point: B-V:0.6 M:4.4 Distance: 1896pc Age: years NGC 188

21. Measuring a distant cluster NGC 2682 0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 B-V

22. Measuring a distant cluster NGC 2682 2: Turnoff point: B-V:0.542 M:4.0 Distance: 870.96pc Age: years

23. Measuring a distant cluster NGC 4590

24. Measuring a distant cluster NGC 4590 Turnoff point:B-V:0.454 M:3.5 Distance: 13085.79 Age: years

25. Measuring a distant cluster NGC 6397

26. Measuring a distant cluster NGC 6397 Turnoff point: B-V:0.596 M:4.4 Distance: 2210pc Age: years

27. Measuring a distant cluster Possible errors……

28. Measuring Hubble constant The apparent magnitude of a supernova is given by Since ( in which L is the luminosity of a type Ia supernova and D is the distance between it and us ) We can find that With absolute magnitude M (observed at a distance of 10pc ), we can find that

29. Measuring Hubble constant Set We got Considering errors in both m’ and D Since, We got With the data of m’ and its error △ m’, we can compute D and its error △ D.

30. Measuring Hubble constant The Hubble constant is defined by We can do linear regression to z (with error △ z) and D (with error △ D) to estimate their ratio.

31. Measuring Hubble constant A) Simply using EIV (Errors-in-variable) model (Functional) #Regression through the origin We got Besides

32. Measuring Hubble constant Use MLE (Maximum Likelihood Estimator) to estimate β

33. Measuring Hubble constant First we try to figure out Set We got

34. Measuring Hubble constant Then we can compute β Set

35. Measuring Hubble constant

36. To get our, we have to estimate first. Here we got two methods : a)Using △ z and △ D to get and b) After that, we can get our (β).

37. Measuring Hubble constant Using plan b), we got Too small !!!

38. Measuring Hubble constant B) Sort the supernovae by their distance D

39. Measuring Hubble constant As we see from the graph, the dots are not exactly on a line, but on a curve. As the distance increases, the slope decreases a little. It takes a while for the light of a distant star to travel to earth, so when we look at a farther star, we are actually looking into a more ancient time of the universe. Then we know that the change in the slope by distance actually means the change in Hubble constant by time. The universe is not expanding at a fixed rate, its expansion is accelerating!

40. Measuring Hubble constant If we sort all the supernovae by their D, and define (Which means we do linear regression to the first supernovae.) …… Then we can see the decrease in clearly. (which means the increase in Hubble constant.)

41. Measuring Hubble constant C) Find a more accurate Hubble constant As we see from the graph, at about the point, z = 0.4, the slope changes apparently. So if we compute the supernovae with distance less than,we can get a more accurate Hubble constant. And here we got the Hubble constant Seems more pleasant now!

42. Measuring Hubble constant So the Hubble constant we got is H o =63.7216

43. Thank you !