Download Measuring a distant cluster

Measuring a distant cluster
Measuring the earth
Going up!
M em
mg  G 2
Re
M em
mg1  G
2
( Re  h)
g ( R  h)

2
g1
R
2
Measuring the earth
• Still going up, but look far!
( Re  h) cos   Re
R   D
D
( Re  h) cos  Re
R
Measuring the earth
• Look up!
d 2  r 2  (D  r )2
M eM m
2 2
G
 M m ( ) (R  D  r)
2
( R  r  D)
Tm
r  ( D  r ) sin

2
D12  r 2  ( D  r ) 2
Measuring the earth
m  5.9 10 kg
24
Re  6.37110 m
6
Measuring the solar system
Measuring the solar system
• Measuring angles
DEC       BCE    
2
2
2
AC  AB  BC
2 AB  AC
2
2
2
BC  CE  BE
cos BCE 
2 BC  CE
2
2
2
BE  CE  BC
cos BEC 
2 BE  CE
cos  
Measuring the solar system
• Continuing with Kepler
D
( De  Dm )
k 
3
T
Tm
GM s

k
2
4
2
e
3
e
2
M s  1.989  10 kg
30
Measuring the solar system
rc
distance earth to moon
• Measuring the curvature
distance sun to moon
of the orbit of the moon!
M mm
2 2
m( ) rm  G 2
T
rm
vm2
r
am
M eM m
M sM m
Ftotal  M m a | G
G
rc
rk
G
M sMe
2 2

M
(
) (rc  rk )
2
(rc  rk )
T
|
rk
Measuring the solar system
• But, but, but...
M sM m
M eM m
G 2
G 2
Dsun moon
Dearth moon
• Why the moon doesn't fly away?
• It is!
So what?
Measuring a distant cluster
• Trigonometric parallaxes
Measuring a distant cluster
• Spectroscopic parallaxes
correlation calculation formula
Measuring a distant cluster
Consider two stars of equal luminosity at distances D1 and D2, we can get：
Measuring a distant cluster
Measuring a distant cluster
• NGC 188
Measuring a distant cluster
(mass)
Measuring a distant cluster
NGC 188
Turnoff point: B-V:0.6
M:4.4
Distance: 1896pc
Age:
years
Measuring a distant cluster
• NGC 2682
0
0
5
10
15
20
25
B-V
0.5
1
1.5
2
2.5
2:
Measuring a distant cluster
• NGC 2682
Turnoff point:
B-V:0.542
M:4.0
Distance: 870.96pc
Age:
years
Measuring a distant cluster
• NGC 4590
Measuring a distant cluster
• NGC 4590
Turnoff point:B-V:0.454
M:3.5
Distance: 13085.79
Age:
years
Measuring a distant cluster
• NGC 6397
Measuring a distant cluster
• NGC 6397
Turnoff point: B-V:0.596
M:4.4
Distance: 2210pc
Age:
years
Measuring a distant cluster
Possible errors……
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
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.
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 .
Measuring Hubble constant
• A) Simply using EIV (Errors-in-variable) model
(Functional)
• #Regression through the origin
• We got
• Besides
Measuring Hubble constant
• Use MLE (Maximum Likelihood Estimator) to estimate β
Measuring Hubble constant
• First we try to figure out
• Set
• We got
Measuring Hubble constant
• Then we can compute β
• Set
Measuring Hubble constant
Measuring Hubble constant
• To get our , we have to estimate
• Here we got two methods :
•
a)Using △z and △D to get
b)
• After that, we can get our
(β).
first.
and
Measuring Hubble constant
• Using plan b) , we got
• Too small !!!
Measuring Hubble constant
• B) Sort the supernovae by their distance D
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!
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.)
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!
Measuring Hubble constant
• So the Hubble constant we got is
H =63.7216
o
Thank you !