astrophysics projectpart 1

measuring the radius of earth

by 徐靖 xu jing

• the zenith angle of the sun in summer solstice the zenith angle of the sun varies from plac

e to place(in fact the difference in zenith angle shows the difference in latitude).And if we also know the distance between them, we can calculate the radius of the earth by trigonomitry.

using methods from ancients

- the measurement did by Eratosthenes on Summer Solstice.from the picture we know syene is on Northern Tropic (23°26′14.1″N) and alexandria is at 23°33′26.1″N.we know the distance between alexandria and syene is 843km, so we know the radius of earth:

r = d/θ =843km /7.2 1≈6708.4km(real answer : 6371km )

errows in the above method• the meaning of distance? the distance we need is arc length, but in reality , the distance we measu

re is not exactly arc length,but is the distance travelled. especially when we face mountains(from the difference between modern measurement and ancient ones,it contributes about 700km in the final result)

distance we need

ditance we measure

• what about the atmosphere? we havn't consider the refraction and scattering! (modern measurement

suggests the index of refraction of atmosphere can be up to 1.001,which will contribute up to 0.01 1 in this case.)

• is earth a perfect sphere? if earth is not a perfect sphere , our result will be the local radiu of c

urvature.(contribute only about several km depending on the place)

• when is noon? there can be about up to 12 hours between noon and exact time of s

ummer solstice (contributes up to 0.1275 1 )

• which part of sun do you observe? sun is big in our eyes,when measuring the angle ,where is the "sun"?

which is the angle we need? (from wiki we know sun radius is nearly 0.25 1 from our perspective ,so it contributes up to 0.5 1

solutions

• using planes and helicopters to measure distance.

the distance we measure will be more precise

• choose places near the equator and observe when sun near the zenith.

thus there are less refraction.

• choose lots of places instead of only two. average radiu of curvature will be closer to radius of earth

• "minus" the errow we already know consider the errows and "minus" them from final result.

other method

• we can use doppler shift! earth is rotating,so the relative speed (compared to anything not on th

e earth ) varies. if we observe the same star in differen place/time the spectra of it will be different.by measuring the difference we know linear velocity of the rotation (v).and we know the period is 24h. thus we know the radius of earth.

and we choose sun to observe

r=v/ω=v/(2π/p)=pv/2π

v=(c*Δλ)/2λ

why we choose sun

• big and near so we can focus on the same point on the sun and ignore the rotation

of sun

• high brightness so it is easy to get the spectra

• low relative speed in average if we observe in perihelion or apogee, it is even better.

• having been well observed if we can't observe by ourself , it is easy to get the data we need

other errows & limits

• again, atmosphere.(scattering)

• the radius we calculate is just the distance between the place observer stood and the earth's axis . so we can only calculate the radius in equator .

• high quality data needed

• gravity redshift may cause errow.

furthermore

• using the same method,we can know linear velocity of the rotation of any celestial body

but now we can only measure how broad a peak is on the spectra to know the Δλ(and minus the effect caused by thermal motion,gravity or so).so it required higher quality of data,but we can do it!

• in any place on planet with given radius we can calculate our latitude .

we know the distance from us to the axis!

astrophysics projectpart 2

solar system

by 刘想 liu xiang

11

Cassini’s measurement

He measured the distance between the Earth and Mars in order to calculate the AU.

Cassini chose to look at Mars while in opposition.

Mars looks the largest at that time(we can even see its surface clearly).

Cassini was in Paris (48.8567oN 2.3508oE) and his friend, Richer, was in Cayenne, French Guiana (4.9372oN, 52.3260oW).

Positions differed by an angular distance of 19.7 arcseconds.

Earth

Just like

Mars

A

• Using small angle estimation

• We can find that the distance between Earth and Mars= =75718668.23km

sin

D

A

• We don’t know the period of Mars• Mars reaches opposition once every 779.94 days.• So

• PM=1.88years

• Through Kepler

• RM=1.523AU

1MP

t t

P

2 3( ) ( )M MP R

P R

So 1AU=1.448*108km.

But it is not very accurate.

2 errors:

1.Refraction in the atmosphere

2.they were not observing the same point of Mars.

• Another ways to calculate RE:• 1.known the luminosity of the Sun , set off a

satellite. When the satellite is between the Earth and the

Sun, it spreads. After that, there is a plane. And there are several thermal sensors on it.

We measure the heat it received per second(P).

24

LSP

D

• 2.send laser to Mars while in opposition

• Known the light speed c

• The time t

• RM-RE=0.5ct

An alternative method for calculating the distances to things in our solar systems makes use of the

eclipses of Jupiter’s moons, and knowledge of the speed of light.

Sun

RD1

5.2R

D2

The distance between the Jupiter and the Sun=5.2AU

Theoretical period P1

observed period P2

Light speed c

From |P1 -P2|, we know|D1-D2|

astrophysics projectpart 3

measuring stars

by 刘想 liu xiang 徐靖 xu jing

What is brightness

• Apparent magnitude

• Consider two stars of equal luminosity at distances D1 and D2. Find the difference in their apparent magnitude in terms of the distances.

• The difference=5log10D1 -5log10D2

1026.74 2.5log ( )b

mb

☉

what is luminosity

• The definition of absolute magnitude is observed at a distance of 10 pc.

• 1pc=3.262ly

The answer2

10

126.74 2.5log ( ( ) )

10

L AU

L pc

☉

clusters

what we want to know:•age•color•distance•luminosity•mass ......

for those we can observe directly

• color&luminosity which part of cluster do we observe? - the edge

which wavelength do we observe them in? - not blue

why?

i think it is mainly because some clusters are dense( like glubular clusters) so the interstellar medium became sort of like the atmosphere on earth,and it just cause scattering and refraction as well! so if we look at it not on the edge , we will see blue"sky"! we cannot see stars, so we observe the edge and avoid using blue light.

• data from UBV photometric system(V can represents the brightness,B-V referred to its color )

from the main sequence turnoff, we know that M23 is the youngest, while NGC2682 is the oldest one.

but in order to estimate their age accurately, we have to calibrate our axes to transfer them into standard H-Rdiagram

|

table we use...and turnoff points

• then we know the turnoff point is F0(M23) ,F5(NGC2682),G8(NGC188). so,for their turnoff points:M23:m=+10,M=+2.7 => d≈288.4pcage≈ 2.46*10^9 yearsNGC2682:m=+13,M=+3.5 => d≈794.3pcage≈ 4.17*10^9 yearsNGC188:m=+16,M=+5.5 => d≈1258.9pcage≈ 1.55*10^10 years

ERROWS

• main sequence is wide ,so the luminosity we get from the table is not very accurate.

• the brightness can be effected by nearby stars,dusts,gravitational lensing,etc

• the point we choose is kinds of randomly(on the main sequence but can be a little bit lower or higher),which can lead to some errows.

more....

• we measure brightness by "v", but v only stands for brightness in visible light, so we maybe mistaken the brightness of some blue stars as well as red ones

• light of different wavelength being absorbed(by dusts)differently, this may cause mismeasuring of brightness

other method

• by observing rare event in which effects transmit to nearby area in the speed of light

like supernova or so make cluster expanding at c, if we measure the speed o

f expanding by degrees/time(ω), we know the distance: d=c/ω more generaly:

gravity changed at the speed of light!

if we observe several stars orbiting(like binary stars),there will be delays between the time their partner move(or change its mass by some way) and the time they"sense"

the change, if we know the delay (t),we know the distance between them(c*t),then we know the distance between them and us.(as we know their relative speed compared to earth, we can know their orbit )

earth

• observe when we know we form a Isosceles triangle(judge by relative speed)

• d=ct/θ• in fact we can not use stars in stable orbit, we nee

d stars which is "kicked away" ( by a star passing by or so ) suddenly to test the delay. i post the picture to make it easy to understand.

example • we know the disk of our galaxy is not a flat , which is caused by the Large Magellanic Cloud and the dark matter around it . from the picture we see the position predict by the shape of our disk is a little delayed compared to the real position of the Large Magellanic Cloud.

• from the delay we can derive the distance between that galaxy and our galaxy . (though in fact we know the distance earlier than we know the shape of disk,and the pictures are just sketch maps )

pictures from <scientific american 2011.11>from the same article we know it is new to its orbit now so the delay exists.

disadvantages and soluutions

• to observe the delay takes a long time(millions of year in fact)

• we can not observe stars which are in stable position.

• to be realistic we had better observe the delay caused by changing luminosity (supernova can accelerate the forming of small stars, so we know the time when the light get there) or so.

astrophysics projectpart 4

the last step of the ladder --hubble constant

by 刘想 liu xiang

According this, H0=53.75141292

To first order one measures the rate at which H is changing by the deceleration parameter

For z<<1 the distance-redshift relation is given to first order by

H0=65.0081873

20

0 0

(1 )

2

c q zczD

H H

So

• q0= -0.601348651870477

thank you for listening!

• by:

徐靖 (xu jing)& 刘想 (liu xiang)