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Physics for Planet Hunters

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Physics for Planet Hunters

Docentföreläsning

C. Clément

1

Hunt for Exoplanets

Planets that are orbiting other stars = Exoplanets

According to http://exoplanet.eu 941 planets discovered.

Many sophisticated techniques exists (fraction of planets discovered%)

Direct imaging(4%)

Doppler shift / Radial velocity (58%)

Transit method(34%)

Other Methods (4%)

planet passes in front of the star=> variation in light

2

Hunt for Exoplanets

Planets that are orbiting other stars = Exoplanets

According to http://exoplanet.eu 941 planets discovered.

Today focus on the following methods

Many sophisticated techniques exists

Direct imaging(4%)

Doppler shift / Radial velocity (58%)

Transit method(34%)

Other Methods (4%)

setup important concepts

leading method

3

Direct Imaging

4

Direct Imaging = Detection of a point source image of the exoplanet

Reflected light from the parent star (in the visible)

Or Through thermal emission (in the infrared)

simulation of 1 Jupiter mass

planet around Sun like star

ISintensity of light from the Star

IPintensity of light from the Planet

- Depends on the composition of the planet
- This ratio is generally very small
- 10-9 for Jupiter-Sun
- 10-10 for Earth-Sun

- Infrared wavelength
- Can be large in very favourable cases
- eg. 1 Jupiter at 0.2 AU

log10(IP/IS)

5

Angular Separation and some Units

Let’s put Earth inside the nearest star system Proxima Centauri at is L = 4.243 ly ~4 1013 km

* 1 light-year ~ 1013 km

1 Astronomical Unit

1 AU = 150 106 km

L = 4 1013 km

Proxima Cent.

Sun

* 1 arcsecond = 1as = 1/3600 degree = 4.84 10-6 rad

So even with 0.1 as we could only see the very nearest neighbours of the Sun

Jupiter is at 5.2 AU

Exoplanets of interest : close to the host star, angular separations 0.1 – 0.5 as “arcseconds”

6

star light

Astronomical Seeing

- Light is refracted through the atmosphere
- Turbulence in the atmosphere
- Refraction index is not uniform nor constant
- Varies on a spatial scale of r0
- the size “stable patches of air”

- 10-20 cm for good conditions
- Changing rapidly with time t0~0.01s
- The image is moving over t0 if one has a
- telescope aperture larger than r0.

7

Point Source

Astronomical Seeing

Resulting angular resolution cannot be better

than 0.3 – 1 as.

Point Spread

Function

PSF(x,y)

After Atmospheric turbulence

Point Spread Function PSF(x,y) used to

describe the response

Either use space telescope or adaptive optics

After long exposure

the seeing disk

8

Adaptive Optics (AO)

DM = Deformable Mirror

9

Adaptive Optics (AO), Wavefront Sensor

D

δθ

array of small lenses “lenslets”

with same focal distance

Each lenslet images object at ∞

Δyis measured with a CCD*.

from f and Δyderive deviation δθ

from normal path.

Measure deformation of the wave front.

Compensate for it using eg. deformable mirror

Shack–Hartmann wavefront sensor.

10

Provide information about δθ

in each region of the aperture

on a time scale comparable

or better than t0.

δθ1

δθ2

δθ3

δθ4

Need to sample the entire aperture of the main mirror each lenslet yields one δθ

Need light source bright enough, guiding star or laser

Information from CCD is processed in real time (~1kHz) and used to modify a deformable mirror.

Compensation is not perfect: depends on nbr. of lenslets and update frequency.

11

Laird Close, CAAO, Steward Observatory

Appears as binary with AO off – Two more stars appear

With the deformable mirror / adaptative optics => go below the 0.3 as limit from seeing

12

Diffraction Limit in Optical Telescopes

λ is the wavelength

pinhole

First minimum located at θ given by

(FK2002 vågrörörelselära och Kvantfysik

destructive interference b/w middle and edge of the hole).

D

Light Intensity

Diffraction pattern from

a single slit opening

Same phenenom in optical system or telescope with aperture D .

The smaller the aperture D => the larger the effect of diffraction

13

Diffraction pattern for a point source

Airy disk

Two close-by point sources

Rayleigh criteria: two points can be separated when the angular separation

is equal or larger than Airy’s disk:

We define Angular resolution of telescope (diffraction limit) this is an angle!

14

Resolution at λ=500 nm

Home telescopeD=0.2mR=600 mas

Hubble space telescope (no seeing) D=2.4 m R=50 mas

Very Large Telescope (with adaptive optics) D=8.2 m R=15 mas

VLT (in interferometer mode)D=130m R~1 mas

(90% loss of light formula above does not apply)

James Webb Space Telescope (JWST, in construction)D=6.5 mR~18 mas

European-Extremely Large Telescope (E-ELT, approved)D=39.3mR~3mas

Without adaptive optics the all ground telescopes would be limited to performance

of a 20 cm telescope.

15

Glare

- A planet at 1 UA (distance Sun-Earth) in Proxima Centauri system corresponds to 0.76 as
- from host star.
- So angularly resolvable
- But there is another problem, the glare from the star.

Constrast depends on star and planet conditions

16

- Criteria for direct imaging of a planet
- Define planet is imaged: Signal / Noise = S/N >5
- Simple model
- Consider diffraction from the star as the only source of background light
- Other sources of background light include
- Sky background
- Scattering from telescope elements (eg. edge of composite mirrors, support structures)
- - Adaptive optics halo contribute to the background
- Consider the diffraction pattern from a 1D slit (eventhough telescope is 2D…)

is the angular separation

b/w planet and star

Planet

observer

Star

17

star light

I0 is light intensity from the star at θ=0

Is is light intensity from the star at θ

we inserted

1D diffraction pattern

approximated

Signal over noise ratio is

Approximate (strict) criteria for planet detection

yields

18

Our simple model yields

- At high θ easier to see faint light
- At low θ planet needs to be a lot brighter
- Can increase sensitivity is I0 is smaller
- Find a way to reduce the star light
- Coronograph

example study for JWST (red dwarf star at 13 ly)

planet-star separation θ [as]

JWST R~18 mas =0.018 as approximately compatible with graph.

Also additional algorithms for background subtraction are usually applied.

Clampin et al. 2001 JWST white paper

http://www.stsci.edu/jwst/doc-archive/white-papers

19

Coronograph

- If we can mask the central star we can limit
- Scattering from telescope elements (eg. edge of composite mirrors)
- - Diffraction light

Lyot Stop: removes diffracted star light

focal mask

Coronograph can remove about 50% of the planet light while 99% of the star light.

increase the S/N ratio by a factor ~50.

20

One of the few directly imaged exoplanet systems HR 8799

HR 8799

HR 8799

Composite image from Keck telescope

No Coronograph

Observed in Infrared

WCS facility at Palomar.

Image obtained with coronograph

System with three planets at 130 ly.

HR 8799 is a young star with young hot planets

21

Conclusion Direct Detection

About 38 planets found with this method so far (0.5 – 10 Jupiter masses mJ)

Extremely challenging, need to detect

Easier in infrared and for young and hot planets

Higher resolution => look closer to the star

Critical to remove stray light and scattering in the telescope, subtraction

Remains very challenging even with planned telescopes (JWST and EELT)

(eg. Jupiter / Earth)

22

Radial Velocity

Doppler Shift Method

23

B barycenter

Principles

S

P

B

mP

mS

r = rSB

Both the planet and the star orbit the barycenter of the planet-star system

Star rotates around B

(much smaller movement than the planet)

observer

24

B barycenter

Principles

S

P

Star moves towards observer

light is blue-shifted

by Doppler effect

B

mP

mS

r = rSB

observer

25

B barycenter

Principles

P

S

mP

mS

r = rSB

B

Star moves away from

observer, light is

red-shifted

observer

26

B barycenter

Principles

P

S

mP

mS

r = rSB

B

Star moves away from

observer, light is

red-shifted

Use blue shift/ red shift to

measure the velocity of the star

versus time

observer

27

What can we learn about the planet?

Extract Periodicity P and star velocity vS(t)

Extract Maximum velocity K

Assume circular orbit for simplicity.

(See M. Perryman for full calculation with general orbits)

Full elliptic orbit can also be used,

vS(t) is no longer an ellipse.

Star periodicity = P

28

What can we learn about the planet?

Extract Periodicity P and star velocity vS(t)

Extract Maximum velocity K

Assume circular orbit for simplicity.

(See M. Perryman for full calculation with general orbits)

Full elliptic orbit can also be used,

vS(t) is no longer an ellipse.

Star periodicity = P

We can use Kepler’s 3rd law applied to a relative orbit of the planet w.r.t. the star

r is the distance star-planet

G is gravitation constant

mS is the mass of the star

We derive r the distance between star and planet from the period

29

What can we learn about the planet? (2)

Stable orbit => centrifugal force in equilibrium with Newton’s force

Centrifugal force on the planet

Newton’s law

mP is the mass of the planet

vP is the velocity of the planet

We derive vP the planet velocity from the star mass and planet-star distance

30

What is the Planet Mass?

B barycenter

S

P

B

mP

mS

r = rSB

We can relate the planet mass with the star mass

We derive the planet mass mP from: star mass and planet and star velocities vP and vS

31

1) Measure the period of the star radial velocity variations

2) Derive the distance between the star the planet (Kepler’s 3rd law)

3) Derive the velocity of the planet

4) Derive the mass of the planet

From radial velocities measurements we can derive the planet mass.

(Note: the stellar mass is know from the star luminosity and spectrum)

32

Inclinded Orbit

What we did was in the special case when the observer is in the planet-star orbital plane.

The radial velocity is entirely towards or away from the observer.

In that case we really measure the full radial speed of the star

observer

33

Minimum Exoplanet Mass

Now the orbital plane is inclined with angle i “inclination”

measured w.r.t. plane perpendicular to line of sight.

The observed star radial velocity is the projection of actual velocity

i

observer

The observed mass is a minimum planet mass

because we do not know the inclination of the orbit.

34

What radial velocities for the stars?

B

Assume circular orbits here

(for complete derivation see M. Perryman Chap2).

Star has a circular orbit around the barycenter of the system.

rS

rS star orbit radius

- Circular orbit
- uniform motion with constant radial velocity K.

We can now derive what radial velocities to expect!

Most easily detectable planets with radial velocities have small Periods and large masses!

35

Examples of stellar radial velocities

Adpated from:

Cumming et al. 1999 Astrophys J 526 890–915

P is the period in year

mJ is the mass of Jupiter

mE is the mass of Earth

mS is the mass of distant star

mSun is the mass of distant star

First discovered exoplanet. M51Pegasi=1.11MSun

Mayor, M. and Queloz, D. (1995). A Jupiter-mass companion to a solar-type star.Nature, 378:355–359.

36

How to we measure the star radial velocity?

Doppler shift!

When the star is moving towards us, its light is shifted to the blue.

When the star is moving away from us, its light is shifted to the red.

Star radial velocity about the star–planet barycentre is given by the

small, systematic Doppler shift in wavelength of the many absorption

lines that make up the star spectrum.

37

Doppler Effect in Classical Physics

observer

observer

source not moving

w.r.t. observer

source moving

w.r.t. observer

c velocity of waves in the medium

vobs velocity of observator in medium

vsource velocity of source in medium

λobs observed wavelength

λem emitted wavelength

see FK2002 vågrörelselära

och Kvantfysik.

Velocities are measured w.r.t. medium

Velocity directions measured w.r.t. an axis eg. connecting source and observed.

38

We are looking at the Doppler shift of the Star, moving away or towards us.

- Take observer as reference system
- vobs=0
- vsource= vS observed radial velocity vS

with

relative

Doppler shift

Special relativistic effects are O(1 m.s-1) and not negligible.

39

Requirements on the spectrograph

for Jupiter we havevr~12.5 m.s-1c =3 108m.s-1

thus need to be able to detect ~4 10-8

absorption lines

Here an example of a very large Doppler shift obtained

with a spectrometer.

In the case of star-planets the lines are shifted

by one part in 108 !

Distant Galaxy

Cluster

Sun

40

Echelle Spectrograph

Use the diffraction grating at high angle

additional

path length Δℓ

θ

i

incoming and outgoing

wave fronts

Difference in path between to rays:

Use configuration with

An echelle grating

i incidence angle

θ diffraction angle

d distance between two lines

The diffraction spectra are located at

d

41

The diffraction spectra are located at

At high values of the order m, the spectras are essentially superposed but slightly shifted.

Coarse grating needed to be able to allow high values of m ( )

If we look at it is diffracted at for

gives

gives

…

the spectrum is diffracted over and over in the same direction

for each order centered on a slightly different wavelength

order m-1

order m

order m+1

superposed

We obtain

multiple superposition of

the spectrum.

43

Remove the superposition with a second grating

perpendicular to the first one

star light

overlaid

diffraction orders

an echelle grating

orders

2nd dispersing element: grating or prism perpendicular to the 1st

wavelengths

44

longestλ

orders

wavelengths

shortestλ

Raw spectrum from comet LINEAR C/1999 S4

45

Resolving Power

Screen/ light detector

Interference+diffraction pattern

leads to maximum fringes separated

by N -2 minima.

d

Position of the minima on the screen

N slits or lines

Width of the maximum fringe

single slit

(same for transmission

and reflection grating)

- With a grating of N lines (or N slits), N-2 minima b/w two max fringes
- the bright fringes become narrower

Relative error onxand λmust be the same

This gives us the resolving power of the grating

46

Resolving power of the grating

Looking back at the grating studied earlier

Assume that the grating is 2cm => N=600

47

Example of spectrum with S/N~1 (Queloz 1995)

Top line is same as next but S/N=40

Although the S/N is low, there are

about 1000 absorption lines that can be

matched, each contains information.

M(v) model of the expected spectra

eg. measured at the beginning

of data taking

Cross-correlation function

Find ε that minimizes this correlation function

48

star light

Spectrograph

+ Camera

- Each invidual line “measures” the Doppler shift with a resolution
- 1000 absorption lines
- Equivalent to measuring the same Doppler shift 1000 times
- Increases precision by

BUT

There could be deformations in the spectrum such that the Model does not fit the data.

Idea= superimpose a well know spectrum on top of the star spectrum

sealed I2 vapor cell

a few cm thick.

49

- The measured absorption spectrum
- from the star is overlad with
- absorption spectrum from I2
- by comparing the measured I2
- absorption lines with those measured
- in the lab
- Equivalent of the Point Spread Function
- for the spectrum.
- Can apply the PSF to the Model before
- extracting radial velocity.

I2 cell temperature controlled

Very stable over years.

Can be calibrated in the lab

with a precision of 1 in 108

Star

this procedure allows to reach

accuracy of 108 on radial velocity.

50

Conclusion Radial Velocity

Most successful method so far (0.5 – 10 Jupiter masses mJ)

Need to measure Doppler shifts of the order of 1 part in 108 / 1m.s-1

Most easily detectable planets have small Periods and large masses

Current technologies Reaching precisions of 1 m.s-1

Remaining Challenge: Earth-like planets: 0.1 m.s-1

Becomes limited by photon statistics!

51

Recommended Reading

M. Perryman, The Exoplanet Handbook, Cambridge Books ISBN: 9780511994852

M. Perryman, Rept. Prog. Phys.63:1209-1272,2000

A. Cumming et al. 1999 ApJ526 890

B.R. Oppenheimer Vol. 3, Chapter 10, p. 157-174, 2003

D. Mawet et al. 2010 ApJ709 53

C. Hanot, PhD Thesis, Universite de Liege, 2011

G. W. Marcy, R.P. Butler, Pub. Astr. Soc. 104: 270-277 1992

D. J. 1976 Applied Optics, 6 11, 1967

S. S. Vogt PASP, 99 621, pp. 1214-1228, 1987

D. Queloz, 167 of IAU Symposium, 221–28, 1995

R. P. Butler et al., PASP, 108, 500-509, 1996

G. W. Marcy, R. P. Butler, R. P. 1998, ARA&A, 36, 57

52

Backup Material

53

Doppler Effect in Special Relativity

with vsource>0 if source is moving away w.r.t. observer

Stars in our galaxy move at speeds O(200km/s)<< speed of light

Hence β<<1.

=> Use a Taylor development

Refraction index of air at the

spectrograph ~< 1m.s-1

To first order same result

as classical approach

54

HR diagram

Luminosity vs.

Color for 22,000

stars from

Hipparcos catalogue

and

1000 from Gliese

catalogue

55

From Perryman book Chapter 2

Which gives for circular orbit and sin(i)=1

same as what we derived from a circular orbit on page 35.

56

Optical Aberration

no aberration

aberration

(barrel)

Image from object an infinity is ideally formed in the focal plane

In case of aberration the image might not form in a single point, or not in the focal plane.

Due to imperfections in the optics, eg. spherical aberration.

Can be wavelength dependent: chromatic aberration.

57

Doppler Effect in Classical Physics

observer

observer

source not moving

w.r.t. observer

source moving

w.r.t. observer

c velocity of waves in the medium

vobs velocity of observator in medium

vsource velocity of source in medium

fobs observed frequency

fem emitted frequency

see FK2002 vågrörelselära

och Kvantfysik.

Velocities are measured w.r.t. medium

Velocity directions measured w.r.t. an axis eg. connecting source and observed.

58

see FK2002 vågrörörelselära

och Kvantfysik.

Wave length

- Take observer as reference system
- vobs=0
- vsource= vr observed radial velocity vr

with

Special relativistic effects are O(m.s-1)

and not negligible.

59

Wave length and frequency

In one period the wave covers one wavelength

Frequency is inverse of the period

For electromagnetic waves in vacuum

60

50ly Sun neighborhood

133 visible stars naked eye

About 10% of the total

Courtesy: R. Powel http://www.atlasoftheuniverse.com/250lys.html

Planet - Star motion

Both the astrometric and Doppler shift method rely on this.

Both the star and the planet are orbiting the barycenter of the star-planet system

Model with a single planet: 2-body system planet (p) and star (s)

ms

barycenter

r(t)

mp

R(t)

R(t)describes the movement of the

barycenter of the planet-star system in the galaxy

Second term corresponds to the movement wrt

the barycenter. If mp = 0 there is no additional

movement.

xp(t)

xs(t)

O

62

Planet - Star motion : Numerical example

Apply to Proxima centauri with hypotethical planet P with

mP = Earth mass = 5.97219 × 1024 kg

mS = 0.243 1030 kg

Assume star-planet distance R= 1 AU = 150 109 m

The maximum displacement rS of the star w.r.t. planet-star barycenter

Proxima Centauri

not to scale!

3700 km

“micro arcseconds”

1’’ = 4.84 10-6 rad

L ~ 4 1013 km

Sun

63

Astrometric Method

Derive the angular observable formula

Size of the effect, illustration with proper motion, parallax and planetary motion

Theoretical limitations (explain how it is obtained!)

Results

64

Angular units

One arcminute = 1’ = 1/60 of a degree

One arcsecond = 1’’ = 1/60 of an arcminute = 4.84 10-6 rad

angle of 1 degree

20cm sin(10)

20 cm

1 m

1.7 cm at 1 meter distance

65

Highly Dispersive Spectrograph / Echelle Spectrograph (1)

Use the diffraction grating at very high angle

Difference in path between to rays:

The diffraction spectra are located at

If we look at it is diffracted at

for

Other orders are diffracted in the same direction

θ

i

gives

gives

…

the spectrum is diffracted over and

over in the same direction, for each

order centered on a somewhat shifted

wavelength

d

an echelle grating

i= incidence angle

θ=diffraction angle

66