Project of astrophysics

Author:Wang Yuning, Sun ningyuan,Dong qifei,Wang zeyu,Zhou haosheng

Brief introduction

Our project is divided into several parts ,but the core of this is the Cosmic Distance Ladder ,which means using the little distance we have measured to get the larger distance ,step by step .And our ultimate goal is to measure the Hubble constant .

Part 1 :the earth

a. since there was no shadow on the surface of the water at noon on the day of summer solstice in Syene, we know that Syene locates on N.

  b. Alexandria is in the north of Syene, so

the shortest shadow appeared only at noon too. And when the sun was at its highest , its zenith anger was , we know that the latitude of Alexandria is N

23.5

7 12 '30 42 '

Part 1 :the earth

C.

Part 1 :the earth

DThere are certainly a lot of sources

of errors ,since the technology in Eratosthenes’s age is not as powerful as that of today. Below are listed some errors .

Part 1 :the earth

1. The sunshine. In the project Eratosthenes assume that the sunshine is parallel ,but in fact it’s not.

2. The longitude . In the question , Eratosthenes assume that Alexandria is exactly in the north of Syene , but actually they are not on the same longitude, so the distance between them is bigger than the one should be counted in our calculation.

Part 1 :the earth

3. The time.

4. The circle .

5. The distance .

6. The well..

Part 1 :the earth

Now let’s see what error magnitude.1.the sunshine 8

6

4

2

2

4

6

8

15 10 5 5 10 15

Part 1 :the earth

The red spot on the upper right corner represents the sun , while the circle represents the earth. We found that the distance between the earth and the sun is 1AU=149600000km, much much much more bigger than 843 km ,and our calculator eliminate the error itself ,so we know that this error can be ignored .The magnitude of this should be nearly 0.

Part 1 :the earth

2 The longitude.The number we want should be the

distance on the longitude ,so the line BC should be parallel to the longitude ,but Eratosthenes take the direct distance into account ,which is not correct .After we check the map online , we find that the vertical distance is about 800 km ,and then we get

So the magnitude of this error is m=6367.65kmR

51 10

Part 1 :the earth

3 The time Now we consider about the time . We

estimate Eratosthenes would take about two days to reach Alenxandria ,so the real zenith angle should be about

So

The magnitude should be

7 6 '

6802.86R km

41 10 m

Part 1 :the earth

4 The circleOK, this error can be

ignored ,too ,since the eccentricity of the earth is 0.0167 ,so it doesn’t matter much .

Part 1 :the earth

5.THE DISTANCE In history ,Eratosthenes got his data in

Hellenic Mile ,measured by 5000 ,and

,so he got 792.5km Assume that the distance between

them is 792.5 km So we get that And we know that the magnitude of

this error is m

1 158.5HellenicMile m

6096.15R km

51 10

Part 1 :the earth

6. The well . Syene is pretty a large ,which is known

as Aswan today ,occupying an area of ,so the distance error ranges from about 40km to 0.If

The error is 40m ,the real distance is 800 km or 880 km, then assume that BC is a straight line and the distance between them is 800m

so we get magnitude of this ranges from to

0

22000km

6367.55R km 51 10

Part 1 :the earth

Then the important part is : How to decrease the impact of these errors?

Here we get some solutions. i. To the error b and e ,we can use modern

technology like electronic map to measure the vertical distance between two cities ,or we can simply choose two cities which share the same longitude.

ii. To the error c, it’s simple to solve: just call one of your friend and ask him to do a favor =W=

iii. To the error f ,we can choose two well that is easy to measure the location.

Part 1 :the earth

Now we want to figure other ways to measure the radius of the earth .Here we figure our three main methods , some are simple ,some are accurate.

Part 1 :the earth

Find a building at the seaside .

( ) cos

cos1

1 cos

SoR R h

hR

2 1

360 86400360 2 1

286400

T T

T T

Part 1 :the earth

Using the moonwith the help of a simple pendulum and

a clock

set a radio wave to the moon and count time

Then we use the moon to get another equation.

21

GMg

R

22

c Td

2

3 2

43

GM

SR d

Part 1 :the earth

Triangles.divide the straight line into a lot of

triangles, and the measurement will be much more accurate .

Part 1 :the earth

EC

DB

NM

A

Part 2:the solar system

A.Mars is the second nearest planet to the earth ,and the second brightest planet. Both first is the Venus .Since these two factors, it was easier for Cassini to observe and get precise data.When opposition happens,the Mars is at the nearest point for the Earth .So it is also easier to observe and get precise data.

Part 2 :the solar system

But ,why not choose Venus?Because the orbit of Venus is closer to the sun than that of earth ,so when the opposition happens, the Venus is between the sun and the earth ,and the sun is too bright for us to see the Venus .So we can’t see it .

Part 2 :the solar system

Guiana

Paris

Part 2 :the solar system

a M

O

B

A

earth

Mars

75

0.8057 64005.38 10

9.550829518 10ABD km

D kmrad

Part 2 :the solar system

Part 2 :the solar system

D1. the assumption 2. when the opposition happens , it’s

not sure that whether Paris and Guiana is just face to the Mars ,and this contribute to the majority of the error .

3. The period4. the measurement of angular

a

M

O

B

A

earth

Mars

Part 2 :the solar system

Now by modern technology ,we get the new data.

So we get that

=56000000D km =780T days

8

8

=1.496 10

2.279 10Mars

AU km

D km

Part 2 :the solar system

Use the Sidereal Period of the earth and the measurement of the earth’s radius , we can get the answer ,too .

2

3 2

8

4

1.495 10

GM

D S

D km

Part 2 :the solar system

FUsing Kepler’s law 3 3

2 2J

J

D DK

T T

5.2JD AU

Part 2 :the solar system G

In our opinion ,the first method is more accurate ,since the data is fewer and the calculation is the main part , so the error is less.

b

a

d2

d1

1 2d d c T

Part 3 :the stars

A 2 21 1 2 2

21 2

22 1

11

22

22 1 1

1 2 21 2 2

=4 4

26.74 2.5lg( )

26.74 2.5lg( )

2.5lg( ) 2.5lg( ) 5lg( )

L D b D b

b D

b D

bm

b

bm

b

b D Dm m

b D D

Part 3 :the stars

Part 3 :the stars

C The age of the globular cluster is

pretty old ,stars of which has become cooler than the others .Their light is closer to infrared rather than blue light ,which is improper to be measured by blue light filters .

The age of the stars at the edge of the cluster is younger ,which are also brighter .It means that the apparent magnitudes of these stars are large enough for us to observe .In this way can we decrease the Malmquist bias.

Part 3 :the stars

D After processing a lot of data ,we

get two pictures of NGC 2682 and NGC188.

Part 3 :the stars

NGC 2682

Part 3 :the stars

Part 3 :the stars

The turnoff is labeled by blue line .the apparent magnitude is 12.65,the type of it is F8.

Part 3 :the stars

And this is the data of NGC 188

Part 3 :the stars

Part 3 :the stars

This is a bit messy ,so let’s make it clearer.

Part 3 :the stars

The turnoff is labeled by blue line .the apparent magnitude is 15.086,the type of it is G5

Besides the main sequence stars ,we also observe Giants and white dwarves in both plots .

Compare the two plots ,we can conclude that NGC188 is older ,since the turnoff of it is on the upper left of that of NGC 2682

Part 3 :the stars

D Find the point of intersection of K0

and main sequence line of each plot , identifying that sites , and calculate the distance .

15

2682 : 15.376

5.59

5lg10

10 906app abs

apparent v

absolute

apparent absolute

m M

NGC m b

M

Dm M

pc

D pc

Part 3 :the stars

and

15

188 : 17.138

5.59

5lg10

10 2040app abs

apparent v

absolute

apparent absolute

m M

NGC m b

M

Dm M

pc

D pc

Part 3 :the stars

E

3

-2682

11.10 10 arcsec

NGC

PD

：

4

-188

14.90 10 arcsec

NGC

PD

：

Part 3 :the stars

F The data on the website is : NGC-2682:1.05mas, the error rate

is 4.5% NGC-188:Not found the data .

Part 3 :the stars

GOther method :

1.Anniversary of the parallax method

2.Using

And this lead to our Part 4

0= zcD H

Part 4 :Hubble constant

A

0= 42kmH k c s Mps

Part 4 :Hubble constant

B

Part 4 :Hubble constant

C

Part 4 :Hubble constant

And

00

0

12

cy

H

ck q

H

0=65 , 0.1977kmH qs Mps

Part 4 :Hubble constant

But here we find a question : the data doesn’t match the image very much .We notice that in 4A some points are far below the line .So in order to decrease the error ,we try to eliminate these points .

Part 4 :Hubble constant

Part 4 :Hubble constant

So we try to eliminate the these possible wrong data ,and get this

Part 4 :Hubble constant

And the hubble constant is

Much closer ?

0=60kmH s Mps

Part 5:correct the Hubble constantBut we get the luminosity from the H-R

diagram, which is extremely inaccurate. Now we use a new method to determine the Hubble constant to 5% precision f∼rom a refurbished distance ladder. Here we go!

Project of astrophysics ——Calibrations of Hubble Constant

Author:Sun ningyuan,Dong qifei,Wang Yuning,Wang zeyu,Zhou haosheng

Part 1:Theory

We calibrate the luminosity by producing a distance ratio using 238 Cepheids from NGC 3021 、 NGC1309 、 NGC 3370. Cepheids have P-L relation as a character : P-L line is a straight line. And the verticaldistance between the P-L line from different galaxies act as the distance ratio.

Part 1:Theory

We get the shortest distance by the a geometric mean : maser distance .

Combining the distance we got and the distance ratios we have calculated. We can get all the distances (of the Cepheids).

As the Supernova and the Cepheids are in one galaxy, we can assume that they are in the same distances from us.

Now we get the distance of the Supernova. Combinig it with the Doppler redshift z ,we cancalculate the Hubble Constant——more accurately!

Part 2:Calculation Step 1Step 1 Settling DataWe get the whole data from a Paper.

Part 2:Calculation Step 1

<V> ——apparent magnitude from visible light filters.Define =<V>

<I> ——apparent magnitude from infrared light filters. Define =<I>

vm

Im

Part 2:Calculation Step 1 Sandage & Tammann (2008) the slope of the

Cepheid P -L relation is sensitive to chemical composition

Solar-metallicity Cepheids used in the distance scale should be calibrated with Galactic Cepheids.

This conclusion could have important consequences for the determination of the distance scale via the LMC. Tammann et al. (2003) used a mixture of BaadeBecker-Wesselink and cluster-based distance estimates to Galactic Cepheids to calibrate the V -band and I-band P -L relations which should then be applicable to the solar-metallicity Cepheids in SN Ia hosts.

Part 2:Calculation Step 1 Corrections for the extinction of

Cepheids in SN hosts are subsequently made by the use of two colors and a Galactic reddening law (Saha et al. 2006). This is equivalent to the use of a “Wesenheit reddening-free” mean magnitude, , defined by Madore (1982) as :

= − R( − ) where R ≡ AV /(AV − AI)≈ 2.5

wm

wm Imvm vm

Part 2:Calculation Step 1We calculate the and the log of

the periods.wm

Part 2:Calculation Step 2Step 2 ： Analysing Data = logP + so we get these points drawn: (logP,

) Then we find a line to fit the plot .

NGC 3370

wa wbwmwm

Part 2:Calculation Step 2

NGC 3021

NGC 1309

Part 2:Calculation Step 2Here are the three lines we create:

NGC 3370 : y=-3.010x+29.05

NGC 3021 : y=-2.9215x+29.154

NGC 1309 : y=-2.6664x+29.264

Part 2:Calculation Step 2 Now we calculate the vertical distances, as

they aren't completely parrallel, we derived

the intervals of each two galaxies.NGC 3021 & NGC1309: (0.446987 , 0.6156082)NGC1309 & NGC 3370 (0.6678956, o.8981076)NGC 3021 & NGC 3370 (o.2o666,0.279407)

Part 2:Calculation Step 3 Step 3: follow-up work The intervals above stand for the differe

nces of apparent magnitudes. We can calculate the distance ratios of each two galaxies from them .

As we have said in Part 1, we can use maser distance to get one of the distances, of which we can't get the accurate data.

15

2

10m D

D

Part 2:Calculation Step 3 Then we get all the distances. Together with Doppler redshift z , use this brilliant formula

we can get: the accurate Hubble Constant!

0H Dz

c

Thank you for watching

Enjoy the stars !