Exoplanets

Question 1

Definition of a planet.

Answer 1

Within solar system, planet must:
I. Be in orbit around the Sun
II. Have sufficient mass to be almost spherical, implying hydrostatic equilibrium
III. Have cleared its orbital neighbourhood; it must be the gravitationally dominant body in the vicinity of its orbit.

Extrasolar, planets if below limiting mass and orbit stars or stellar remnants (formation doesnt matter). If free-floating, “sub-brown dwarfs” (or name thats most appropriate)

Max mass: Minimum mass necessary to initiate thermonuclear fusion of deuterium in core (brown dwarf). ~13 Mⱼᵤₚ / 0.012 M⊙
Sub-stellar objects with true masses above limiting mass are “brown dwarfs”, no matter how they formed nor where they are located.

Question 2

Definition of a dwarf planet

Answer 2

Within solar system, dwarf planet must:
I. Be in orbit around the Sun
II. Have sufficient mass to be almost spherical, implying hydrostatic equilibrium
III. NOT have cleared its orbital neighbourhood; it may not be the gravitationally dominant body in the vicinity of its orbit.

Ceres, Pluto, Haumea, Makemake, and Eris

Question 3

What is hydrostatic equilibrium?

Answer 3

Gradient in pressure (p) can be defined by the gravitational acceleration (g) and the density (⍴), where g can be written as the gradient in the potential.

∇p = ⍴g = -⍴∇φ

For this to be the case, the gradients needs to be parallel, then the object would be spherical.

Question 4

Kepler’s laws

Answer 4

1) The orbit of a planet is an ellipse with the Sun at one of the two foci.

2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Question 5

The definition and meaning of parameters which describe a planet’s orbit

Answer 5

Taking (usually) the star as the centre.

Two planes:
Reference plane, in exoplanets this is usually perpendicular to observer, ie flat surface on the sky. Established using the reference body (star) and some reference direction (celestial north).
Orbital plane, plane that the object orbits in.

Six elements:

Defining the shape and size of the ellipse
* Eccentricity (e) — shape of the ellipse, describing its elongation compared to a circle.
* Semi-major axis (a) — half distance between apoapsis and periapsis.

Defining orientation of the orbital plane
* Inclination (i) — vertical tilt of the ellipse w.r.t. reference plane, measured at the ascending node (where orbit passes upward through reference plane). Measured from reference to orbital planes, angle i.
* Longitude of the ascending node (Ω) — orients ascending node of the ellipse (☊) w.r.t. the reference frame’s reference direction (♈︎). Measured in the reference plane, angle Ω.

Defining location of celestial body
* Argument of periapsis (ω) — defines orientation of the ellipse in the orbital plane, angle measured from ascending node to the periapsis (closest point from satellite to primary body), angle ω.
* True anomaly (ν) — position of orbiting body along ellipse at epoch (t₀), angle ν from the periapsis.

See Fig(1.21)

Question 6

Barycentric view, star orbital parameters in terms of planetary ones

Answer 6

e⋆ = eₚ
a⋆ = aₚMₚ/M⋆
i⋆ = iₚ
Ω⋆ = Ωₚ
ω⋆ = ωₚ + π
ν⋆ = νₚ

Question 7

Eccentricity equation

Answer 7

b² = (1-e²)a²

Question 8

Position of a planet with respect to the system barycentre/focus

Answer 8

r = a(1-e²)/1+ecosν
where angle ν is the true anomaly, and r is distance from barycentre/focus to the planet

OR

eccentric anomaly E, where E is angle measured from the centre of the ellipse to theoretical circle, and planet is vertically below (where RA triangle would pass through ellipse)
cosν(t) = cosE(t)-e/1-ecosE(t)

OR

mean anomaly M
M(t) = 2π/P (t-tₚ) = E(t) - esinE(t)
tₚ is time at periapsis

Question 9

Derive the planet equilibrium temperature

Answer 9

Mostly determined by light incoming from star.

L⋆ = 4πR⋆²σT⋆⁴

Luminosity recieved by planet:
Lᶦⁿᶜᶦᵈ = L⋆(πRₚ²/4πa²)

Amount that remains is factor of 1 - Albedo:
Lₚ,ᵢₙ = (1−A)Lᶦⁿᶜᶦᵈ = L⋆(1−A)(πRₚ²/4πa²)

Equilibrium:
Lₚ,ᵢₙ = Lₚ,ₒᵤₜ
Lₚ,ₒᵤₜ = 4πRₚ²σTᵉᑫ⁴

Solve for effective temperature:
Tᵉᑫ = (R⋆/2a)¹’²(1−A)¹’⁴T⋆

Tᵉᑫ is purely to incident energy received from host star.
It ignores:
- feedback from the planet’s
atmosphere
- possible sources of internal
heating, incl. tidal heating
effects
- non-blackbody behaviour of
planets and host stars

Question 10

Using a simple model atmosphere, explain why Earth’s temperature is higher than its equilibrium temperature

Answer 10

Simple greenhouse model.

S: from star, high energy solar radiant emittance, splits into:

• Fᵣ = AS: Solar radiation reflected by bond albedo A.
• Temp = Tᵉᑫ at top of atmosphere
• Fᵢ = (1−A)S: Net high energy incident solar radiation
• Temp = Tₛ at surface

From Earth’s surface
Fₛ = σTₛ⁴: Low energy radiation from Earth’s surface
Fₐ = εσTᵉᑫ⁴: Atmosphere (absorptivity = emissivity = ε) heated by low energy radiation from surface, absorbs ε and emits (1−ε) back into space.

Atmosphere also produces its own thermal emission with emissivity ε, half of which radiates into space and half radiates back to the surface.

In equilibrium:

Top of atmosphere
Incoming solar = outgoing surface + atmosphere
Fᵢ = (1−ε)Fₛ+Fₐ
⇒ (1−A)S = (1−ε)σTₛ⁴ + εσTᵉᑫ⁴

Surface
Incoming solar nd atmosphere = outgoiing surface
Fᵢ + Fₐ = Fₛ
⇒ (1−A)S + εσTᵉᑫ⁴ = σTₛ⁴

This implies:
εσTₛ⁴ = 2εσTᵉᑫ⁴
⇒ Tₛ = 2¹’⁴Tᵉᑫ

Surface 19% warmer than equilibrium.

Earth A ~0.3, implies Earth surface temp ~300K (actual is 290K)

Question 11

Earth runaway greenhouse effect possible causes

Answer 11

Sources of positive feedback:
* Melting ice caps leading to less reflected radiation (lowering Bond albedo and raising fraction of incident radiation to the surface)
* Heating oceans causes increased water vapour in the
atmosphere (increasing opacity to thermal radiation re-emitted
back from the surface).

Not strong enough (yet!) for runaway effect on Earth, though it is thought that this may have occurred in the past on Venus.

Question 12

Photon noise limit and S/N ratio

Answer 12

n is mean number of photons expected, k is number observed.
Poisson, var = mean
σ² = n

σ = √n’ ≈ √k’, photon noise limit

Quality of observation determined by signal-to-noise ratio:
S/N = k/σ ≈ k/√k’ = √k’

Better observation with more photons, signal grows faster than noise.

Question 13

What is the radial velocity (RV) exoplanet detection technique?

Answer 13

A star-planet system will orbit around the barycentre, the common centre of mass.
The motion of the host star about the barycentre can be detected by looking for a Doppler shift in the absorption lines of the host star’s spectrum.
We can use the relativistic Doppler formula to calculate the
observed wavelength from the emitted wavelength for a given
radial velocity.
The orbit of the host star will cause a periodic variation in radial velocity, giving a corresponding periodic variation in observed wavelength

Question 14

Doppler formula

Answer 14

Relativistic:
λᵒᵇˢ = λₑₘ 1+(vᵣ/c)/√1−(vᵣ/c)²’
Non-relativistic:
Δλ/λ = vᵣ/c

Question 15

What is the astrometry exoplanet detection technique?

Answer 15

For planets in near face-on orbits, there may be no detectable RV
signal. Instead you can try to detect a planet by looking for a spatial (astrometric) shift of the host star on the sky in response to the presence of the planet.

Question 16

What is the transit exoplanet detection technique?

Answer 16

Planet passing infront of a star (and star passing in front of planet) reduces total light that reaches Earth. Looking for periodic dips in a lightcurve can show presence of a planet.

Question 17

What is the transit-timing variation (TTV) exoplanet detection technique?

Answer 17

Periodicity of a transit can be affected by several factors:
* gravitational influence of another planet, especially in the case of mean motion orbital resonance
* planet or host star tidal effects – other planets or moons
* transit around a binary star system (circum-binary exoplanet)
* precession of periapsis due to general relativistic effects
* the presence of moons around an exoplanet - exomoons

Question 18

Orbital resonance

Answer 18

Planetary orbits affected by gravitational effects of neighbouring planets, inducing small orbit perturbations.
Frequency of such effects governed by ratio of the planets’ orbits.
Perturbing effects usually happen at random points in orbit so therefore average out over time.
If orbital periods obey a simple integer relation then perturbations happen at the same point in an orbit, the effect can be magnified over time.

eg orbital resonance with Jupiter caused gaps in asteroid belt at integer ratios.

Two bodies of orbital periods P₁ and P₂, obey small-integer relation:
m/P₁ − n/P₂ = 0 (for small int. m,n)
yielding a m:n orbital resonance.

Periodic variation in the transit ephemeris is predicted when a pair of planets have periods that are close to (but not exactly) a mean motion orbital resonance. This period is given by
1/Pᵀᵀⱽ = |m/P₁ − n/P₂|

Where |m-n| = N we say that the resonance is an Nth-order resonance.

Question 19

What is the microlensing exoplanet detection technique?

Answer 19

A foreground compact mass will deflect light from a background. Surface brightness is conserved whilst being lensed, but there is an angular increase in image size. This has a magnifying effect. Multiple lensed images will not be able to be resolved from one another, and so we can only observe their combined brightness. A peak in a lightcurve can be indicative of a celestial body.

Question 20

What is the direct imaging exoplanet detection technique?

Answer 20

Viewing planet directly

Question 21

Radial velocity calculations

Answer 21

Standard results
Dist of star from barycentre at time t:
r(t) = a⋆(1−e²)/[1+ecosν(t)]

Semi-minor axis:
b⋆ = a⋆(1−e²)¹’²

Kepler’s 2nd law in polar coords:
½r²ν̇ = πa⋆b⋆/P

Orbit’s geometry gives star’s displacement from reference plane at time t as:
z(t) = r(t)sin(i)sin[ω⋆+ν(t)]

Radial velocity is the time derivative of z
vᵣ = ż = sin(i)[ṙsin(ω⋆+ν)+rν̇cos(ω⋆+ν)]

Take time derivative of r from standard result:
ṙ = ae(1−e²)ν̇sinν/(1+ecosν)²
= rν̇esinν/(1+ecosν)

Substituting into ż equation, use hella angle formulae:
vᵣ = rνsin(i)/(1+ecosν) [cos(ω⋆+ν) + ecosω⋆]

Use dist of star from barycentre and K2L to simplify:
vᵣ = 2πb⋆/P sin(i)/(1-e²) [cos(ω⋆+ν) + ecosω⋆]

Convert semi-minor to semi-major:
vᵣ = 2πa⋆/P sin(i)/√1-e²’ [cos(ω⋆+ν) + ecosω⋆]

vᵣ(t) = K {cos[ω⋆+ν(t)] + ecosω⋆}

where K = 2πa⋆/P sin(i)/√1-e²’
is the radial velocity semi-amplitude

Shape approximately sinusoidal.

To convert to exoplanet parameters, use
P² = 4π²/G(M⋆+Mₚ) a³
a = a⋆ + aₚ
M⋆a⋆ = Mₚaₚ

P² = 4π²/GM’ a⋆³
with M’ = Mₚ³/(M⋆+Mₚ)²

Use to replace a⋆

Question 22

Astrometry angular displacement calculations

Answer 22

Minimal requirement for astrometric detection is to observe a statistically significant and periodic shift in the centre of light of the host star’s image, relative to a fixed reference frame.
The angular displacement of the star from the planet—star barycentre is:
θ = Mₚ/M⋆+Mₚ a/d ≃ Mₚ/M⋆ a/d
(Mₚ ⟨⟨ M⋆)
where a = aₚ + a⋆ and d is distance to the star from observer.

Converting into milli-arcsec (mas) angular units gives
θ = 1 mas for units Mⱼᵤₚ1AU / M☉︎1pc

Question 23

Astrometry centroid accuracy

Answer 23

Random arrival time of photons means host’s centroid accuracy σ꜀ is governed by Poisson uncertainty in photon rate from the host:
σ꜀ ∝ Nₚₕₒₜₒₙ⁻¹’² ∝ (S/N)⁻¹

It is also limited by the resolving power of the telescope at the observation wavelength λ:
σ꜀ ∝ λ/D
where D is the telescope diameter.

Combining equations, and including the missing proportionality constant (4π):
σ꜀ = λ/4πD (S/N)⁻¹

Question 24

Requirement to observe any transit

Answer 24

Planet with host separation a and orbit inclination i, producing a sky-projected closest approach distance z = a sin(i) between planet and host, requirement to observe any transit is
z = asin(i) ⟨ R⋆ + Rₚ

Full transit, all of planet covers fraction of star.
Grazing transit, fraction of planet covers fraction of star.

Question 25

Full transit magnitude/flux change
(transit depth)

Answer 25

Host star’s flux normally F⋆, during transit it may attain minimum flux determined by maximum fraction of the host’s area covered by the planet:
Fₘᵢₙ = F⋆[1−(Rₚ/R⋆)²]
(assumes host star has a uniform surface brightness distribution across its disk)

ΔMₘₐₓ = −2.5log₁₀(Fₘᵢₙ/F⋆)
= −2.5log₁₀[1−(Rₚ/R⋆)²]

Question 26

Ingress and egress

Answer 26

Ingress inbownd, egress outbound.
‘Corners’ of lightcurve at points where planet just inside/outside of star’s disk.

Ingress/egress time longer for less depth.

t₁₄ time of whole transit
t₂₃ time at min. flux

Question 27

Flux during ingress and egress

Answer 27

Fig(4.21)

Before derivation:
* Let (x,y) origin be at centre of
host.
* y points upwards.
* All distances in units of host
radius (so host radius is unity).
* Position of planet centre is (z,0)
* Chord of intersection between
host and planet has length 2y.
* Position of centre of chord is
(x,0) and the top is (x,y).
* Distance between chord and
planet along x-axis is z−x (-ve
when planet centre is to the left
of chord)
* Planet radius is p (expressed in
units of host star radius)

Assume:
* Planet in circular orbit (e=0)
* Host star has uniformbrightness profile
* Planet has no/ an opaque atmosphere
* Planet flux negligible (no secondary transit)

Equate circles at top of chord:
x² + y² = 1
(z−x)² + y² = p²
⇒ (z−x)² + 1 − x² = p²
Solve for x, then sub for y
x = 1−p²+z²/2z
y = √4z²−(1+z²−p²)² / 4z²’

Overlap area is sum of two lens-shaped areas, Aₕ area of host to right of chord, Aₚ area of planet to left of chord.
Aₕ = difference in circular wedge and triangle
= θ/2 − yx = cos⁻¹(x) − yx
For Aₚ, use z−x instead of x
Aₚ = p²cos⁻¹[(z−x)/p] − y(z−x)

F(t) host flux at time t, F⋆ is flux in absence of transit,
F(t) = [1−f(t)]F⋆
where
f(t) = π⁻¹(Aₚ + Aₕ)
= 1/π{p²cos⁻¹[(z−x)/p] + cos⁻¹(x) − yz}

Question 28

Transit duration

Answer 28

Fig(5.6)

Circular orbit in time Δt, planet sweeps angle
φ = 2sin⁻¹(b/a) = 2πΔt/P

Path half-length across the host disk:
b = √R⋆² − z² ‘ = √R⋆² − a²cos²(i) ‘

rearrange and sub:
Δt = P/π sin⁻¹[√R⋆² − a²cos²(i)’ / a]

This is transit duration for Rₚ ⟨⟨ R⋆ (ie ignoring ingress/egress times)

Question 29

Transit probability

Answer 29

Fig(5.7)

Small planet in a circular orbit about its host casts a shadow which, over an orbit, traces out a band-like shadow region on the celestial sphere.

Probability that this planet will be observed by us as transiting its host the ratio of the area of its
shadow region to the area of the
surface of the celestial sphere.

θ is small and y ⟩⟩ a, so can write
P = 2π(a+y)S/4π(a+y)² = S/2(a+y)
where S = yθ, and 2R⋆ = aθ (R⋆ is host radius)
⇒ S = 2R⋆y/a

Sub in:
P = R⋆/a (y/a+y) ≃ R⋆/a
(y ⟩⟩ a)

Allowing planet of non negligible radius, R⋆ → R⋆ + Rₚ
P ≃ R⋆+Rₚ/a

Question 30

Density from RV and transit methods

Answer 30

Transiting systems are often the target of RV follow-up.
Transit implies inclination must be close to 90°
If RV signal is also detected, get Mₚsin(i) (assuming M⋆ is known). But sin(i) ≃ 1 so we get Mₚ directly.
Transit observations give Rₚ, combination of Rₚ and Mₚ allows the average internal planet density to be determined.

Question 31

Detectability calculations for TTV

Answer 31

Inner planer arrives late for transit due to grav force from trailing outer planet.
Inner planer arrives early for transit due to grav force from leading outer planet.

Question 32

Detecting exomoons

Answer 32

Exomoons induce two basic types of timing effect:
* Transit Timing Variation (TTV) which produces a periodic behaviour of the ingress time
* Transit Duration Variation (TDV) where total duration of transit by the planet changes periodically
* TTV and TDV signals should be out of phase by π/2 if the
exomoon and exoplanet orbits are coplanar. Combination of
TTV and TDV signals exhibiting such a phase difference is a key
signature needed to confirm an exomoon.

Question 33

TTV due to exomoon

Answer 33

Time of ingress is determined by the position of the exoplanet relative to the exoplanet-exomoon barycentre.

Early ingress if exomoon ‘behind’ exoplanet in its orbit.

No change in ingress time if exomoon in line with exoplanet.

Late ingress if exomoon ‘ahead of’ exoplanet in its orbit

Fig(6.4)

Question 34

TDV due to an exomoon

Answer 34

The duration of transit is modified by the orbit effect of the exomoon around the planet host.

Has effect if exomoon in line with exoplanet w.r.t. host star.

Slower transit if exomoon travelling in same direction as exoplanet.

Faster transit if exomoon travelling in opposite direction to exoplanet.

Fig(6.5)

Question 35

Amplitude of exomoon TDV

Answer 35

Exoplanet mass Mₚ in circular orbit of radius aₚ, host star mass M⋆. Exomoon mass Mₘ in coplanar circular orbit of radius aₘ.

Change in transit induced will have maximum amplitude δt(TDV) relative to transit duration Δt. For edge-on transit, (i=90°):
δt(TDV)/Δt = δvₚ/vₚ
where vₚ is planet orbital speed and δvₚ is amplitude of speed variation due to exomoon.

Assuming Mₚ ⟩⟩ Mₘ:
vₚ = √GM⋆/aₚ’
δvₚ = vₘ(Mₘ/Mₚ) = √GMₚ/aₘ’ (Mₘ/Mₚ)

Substitute in:
δt(TDV)/Δt = δvₚ/vₚ
= √aₚMₘ²/aₘMₚM⋆’

Question 36

Derive the Einstein radius

Answer 36

In the simplest case, a foreground compact mass will deflect light from a background by an angle (in radians):
α = 4GM/c²b
where M mass of the deflector body (“lens”) and b is the closest approach of the light path to the lens. Gravitationally lensed background star is the “source”.

See Fig(6.16) for diagram of point source point lens system.
Small angle approx.: tanθ ≃ θ
θDₛ = θₛDₛ + αDᴸₛ

Substitute for α and note that b = θDᴸ:
θDₛ = θₛDₛ + 4GMDᴸₛ/c²b
= θₛDₛ + 4GMDᴸₛ/c²θDᴸ

For special case θₛ = 0 (source directly behind lens):
θDₛ = 4GMDᴸₛ/c²θDᴸ

Case where θₛ = 0 defines angular Einstein radius:
θᴱ = √4GMDᴸₛ/c²DₛDᴸ’

The corresponding physical Einstein radius (measured in the lens plane) is
Rᴱ = θᴱDᴸ = √4GMDᴸDᴸₛ/c²Dₛ’

Physical interpretation:
Source located exactly behind the lens will produce images which merge to form a ring with angular radius θᴱ

Question 37

Lens equation for image locations

Answer 37

For a given deflection angle we can compute how many images are produced and where they are located. Note that:
α = 4GM/c²b = 4GM/c²θDᴸ
= θᴱ²/θ Dₛ/Dᴸₛ

Can substitute θᴱ for α to get:
θ² =θₛθ+ θᴱ²
This is the lens equation.

Quadratic has two solutions for the image positions (θ₁,θ₂):
θ = θ₁,₂ = θₛ/2 ± ½√θₛ² + 4θᴱ² ‘

Negative value for θ₂ indicates image is formed on the opposite side of the the lens to the true source position, and is inverted.

Images are always co-aligned with the lens and source. One image forms outside the Einstein radius, the other inside. As the source moves away from the
lens, θ₁→θₛ and θ₂→0. First image
becomes coincident with true source location, and the second image becomes coincident with lens position (“hides” behind it).

See Fig(6.22)

Question 38

Microlensing image magnification

Answer 38

Magnification A = (area of image)/(area of source):
A₁ = |θ₁/θₛ dθ₁/dθₛ|

Normalise angles to Einstein radius θᴱ. Define:
u = θₛ/θᴱ
u₁ = |θ₁|/θᴱ
where u is the impact parameter

Rewrite lens eq:
θ₁,₂ = θₛ/2 ± ½√θₛ² + 4θᴱ² ‘
u₁ = ½(u² + 4)¹’² + ½u
u₂ = −½(u² + 4)¹’² + ½u

Magnification factor becomes:
A₁ = u₁/u du₁/du
A₂ = u₂/u du₂/du

du₁/du = u/2(u²+4)¹’² + ½
du₂/du = |−u/2(u²+4)¹’² + ½| = |u/2(u²+4)¹’² − ½|

A₁ = u²+2/2u(u²+4)¹’² + ½
A₂ = u²+2/2u(u²+4)¹’² − ½

Since we cannot resolve individual images, we observe an overall magnification:
A = A₁ + A₂ = u²+2/u(u²+4)¹’²

As u → 0, magnification diverges

Question 39

Einstein radius crossing time

Answer 39

Uniform relative motion between the lens and source:
S(t)² = S₀² + Vₜ²(t−t₀)²
where Vₜ is the transverse speed

Define u = |θ|/ θᴱ
S/Rᴱ = θₛ/θᴱ = u

u(t)² = u₀² + (t−t₀/tᴱ)²
where u₀ = S₀/Rᴱ is the minimum impact parameter, and
tᴱ = Rᴱ/Vₜ = θᴱ/μᵣₑₗ
is the Einstein radius crossing time.
μᵣₑₗ is the relative proper motion (angular speed)

Question 40

Microlensing optical depth

Answer 40

Physically, optical depth gives number of ongoing microlensing events at a given time for a random source star at
distance Dₛ.
Scales with the lens mass density (not with number density!) and is independent of lens mass M.

Lenses with mass M and number density distribution n(Dᴸ). Differential mass density of lenses in interval Dᴸ, Dᴸ+dDᴸ:
dΣ(Dᴸ) = n(Dᴸ)MdDᴸ = ⍴(Dᴸ)dDᴸ

Surface mass density within an Einstein radius is
Σ(lens) = M/πRᴱ²

Microlensing optical depth for star at Dₛ is:
τ = ∫Dₛ dΣ/Σ(lens)
= ∫Dₛ πRᴱ²/M ⍴(Dᴸ) dDᴸ
= 4πG/c²Dₛ ∫Dₛ DᴸDᴸₛ⍴(Dᴸ) dDᴸ

Question 41

Microlensing rate

Answer 41

Time inside microlensing tube:
T = 2tᴱ√1−u₀² ‘

P(u₀) = const. (all values equally likely), so average T is:
⟨T⟩ = 2⟨tᴱ⟩ ∫₀¹ √1−u₀² ‘ du₀

Sub in u₀ = sinθ:
⟨T⟩ = ⟨tᴱ⟩ cosθsinθ + θ
= π/2 ⟨tᴱ⟩

Microlensing rate Γ = number of ongoing events / average time in microlensing tube
Γ = τ/⟨T⟩ = 2/π τ/⟨tᴱ⟩

Question 42

Lens mass

Answer 42

M = θᴱ²/κπᵣₑₗ

κ = 4G/c²(1AU)
πᵣₑₗ = (1AU)(Dᴸ⁻¹−Dₛ⁻¹)

Question 43

Ratio of total planet flux to star flux

Answer 43

Planet flux is sum of intrinsic planet flux Fᵢ from intermal energy sources, and reflected flux of host star Fᵣ.
At wavelength λ:
Fₚ(λ,φ)/F⋆(λ) = Fᵢ(λ)+Fᵣ(λ,φ)/F⋆(λ)
= Fᵢ(λ)/F⋆(λ) + G(λ)P(φ)(Rₚ/a)²
where G is the geometric albedo and P is a phase function.

Phase function gives fraction of the planet disk illuminated by the host star as viewed by the observer. Phase function depends on the position angle Φ of the planet in its orbit around the host star.

Question 44

Geometric albedo

Answer 44

The geometric albedo of a planet is the ratio of the brightness of a planet viewed from the direction of illumination compared to that of a flat Lambertian disk with the same cross-sectional area. The
geometric albedo is wavelength dependent.

A Lambertian surface diffusively reflects 100% of incident radiation such that each element of the surface is an isotropic emitter.

Unlike the Bond albedo, the geometric albedo can exceed unity, such as for planets that exhibit specular reflection due to oceans or ice. In our Solar System the icy Saturnian moon Enceladus has a Bond albedo of 0.99 but a visual geometric albedo of 1.4.

Question 45

Radiation from Kelvin-Helmholtz contraction

Answer 45

Consider gravitational potential of a shell of matter surrounding mass M.
For a non-rotating system in virial equilibrum, kinetic energy (T) is related to gravitational potential energy (U) by:
2T = U

Up to half of the potential energy is available to radiate as thermal energy as a planet cools and contracts.

Potential energy of the shell:
dU(r) = −GM(⟨r)/r dM(r)

Assuming a power-law density profile:
⍴ = Krⁿ

Differential mass of the shell is therefore:
dM(r) = 4π⍴(r)r² dr
= 4πKr²⁺ⁿ dr

Interior mass out to r:
M(⟨r) = ∫ʳ dM(r’) = 4πK/3+n r³⁺ⁿ
(n ⟩ −3)

Constant K can be obtained from total planet mass Mₚ and radius Rₚ:
Mₚ = 4πK/3+n Rₚ³⁺ⁿ
⇒ K = 3+n/4π MₚRₚ⁻⁽³⁺ⁿ⁾

Combining:
dU = −16π²GK²/3+n r²⁽²⁺ⁿ⁾ dr

Integrate from initial collapse (infinity) to the current planet radius to obtain total potential energy which can be radiated away:
U = −16π²GK²/3+n ∫(∞→Rₚ) r²⁽²⁺ⁿ⁾ dr
= −16π²GK²/(5+2n)(3+n) Rₚ⁵⁺²ⁿ
(−3 ⟨ n ⟨ −5/2)
-ve sign denotes energy loss.

Substitute for K and apply Virial theorem, get thermal energy due to contraction:
T = U/2 = −(3+n/10+4n) GMₚ²/Rₚ

This mechanism is significant for Jupiter which radiates more energy than it receives from the Sun.

Question 46

Pulsar timing method and calculation

Answer 46

Strong magnetic field lines confine pulsar emission to highly
collimated outflows. Misalignment between pulsar spin axis and axis defined by emission outflow produces pulsed emission for lines of sight which periodically intercept the beamed emission.
Beamed emission is observable at radio wavelengths, allowing pulsar periods to be measured. Some millisecond pulsars have rotation periods which rival precision of atomic clocks.

If pulsar hosts a planet, pulsar will orbit the pulsar-planet barycentre. Its orbital motion induces a modulation in the arrival time of pulsed emission due to a change in the path length of the photons between the observer and pulsar:
Δτ = 1/c Mₚ/M⋆ asin(i)
where i is the inclination
of the planet’s orbit.

Question 47

Pros/cons/limitations for RV

Answer 47

• Able to be done on Earth, majority of ground-based detections have used the RV method.
• Amplitude of the RV effect is sensitive to i. Face-on orbit (i=0°), no RV signal.
• The signal is stronger for larger a
• Formula for K not valid in relativistic limit, but not of concern in most extrasolar planet systems.
• Stellar activity can mimic a positive planetary RV signal

Question 48

RV spectrograph parameters

Answer 48

Velocity precision governed by:
* Number of lines that can be measured. Wider spectral range, more lines.
RV precision:
σᵣᵥ ∝ ΔvᵣNₗᵢₙₑₛ⁻¹’² ∝ ΔvᵣBλ⁻¹’²
where Δvᵣ is velocity precision from single line and spectrum wavelength range is Bλ
* Spectral resolution R at given wavelength λ is:
R = λ/Δλ = mL/d = c/Δvᵣ
⇒ σᵣᵥ ∝ R⁻¹
where m is diffraction order, L is total width of the grating and
d is grating spacing.
* Signal-to-noise ratio (S/N) of host star flux within each detector channel (pixel), which determines whether lines can be
identified and measured. At Poisson limit this gives:
σᵣᵥ ∝ Nₚₕₒₜₒₙₛ⁻¹’² = (S/N)⁻¹

Overall RV precision:
σᵣᵥ ∝ R⁻¹(S/N)⁻¹Bλ⁻¹’²

Question 49

Echelle spectrograph

Answer 49

Reflection grating, maximises R (spectral resolution). Uses blazed grating optimised to high spectral orders m. Fig(3.14)

Grating equation, maxima where
d(sinα+sinβ) = mλ

Blaze angle θᴮ is defined by angle of grating facets where α = β, which maximises throughput to order m. This gives
θᴮ = α = β = arcsin(mλ/2d)

Optimising for high m requires large θᴮ. Since high orders typically have over-lapping spectra, a cross disperser is used to separate them.

Question 50

Limitations for astrometry

Answer 50

Ground based, seeing effects observations.
σ꜀ ≃ 3 mas
Corresponds to 0.03AU motion amplitude for host 10pc away. VERY BAD

Space-based, no seeing so theoretical limit can be approached, σ꜀ = λ/4πD (S/N)⁻¹
GAIA can measure σ꜀ = 8 micro-arcsec (µas) for bright stars.
With µas positional accuracy, must correct for:
* Gravitational deflection of light by the Sun.
* Special relativistic corrections in apparent position of objects due to Earth’s motion Solar System barycentre (aberration). Observer’s velocity needs to be known with an accuracy of ~0.001 m/s to reach µas accuracy!
* Perspective acceleration, 3D motion of stars in space leads to
change in angular motion on the sky as star moves nearer or further away. This effect is significant at the µas level out to tens of pc.

Question 51

Limitations and complications for transit method

Answer 51

Uniform source, transit inclination and depth are easy to decouple with reasonable data coverage.
However, In optical bands hosts show limb-darkening, radially decreasing brightness. Makes calculations more difficult, produces symmetric curvature to the transit dip.

Need RV confirmation for all systems.

Basic transit photometry provides:
* radius of planet in units of the host radius
* orbital period
* inclination angle
* eccentricity (sometimes)
But alone it CANNOT determine the planet mass

Can also get false positives from an unresolved distant eclipsing binary system contaminating the flux.

Question 52

Precision of transit measurements

Answer 52

σₜ for a single transit is governed by detecting a difference in the number of photons ΔN received during transit and out of transit. Since photon error is governed by Poisson statistics we have
σₜ ∝ (ΔN)⁻¹’² ∝ (F⋆−Fₘᵢₙ)⁻¹’² ∝ Rₚ⁻¹ ∝ Mₚ⁻¹’³

Precision can be improved by observing multiple transits. For a fixed total observing time T and a planet with period P we can observe n = T/P transits. We can expect the precision to improve as
σₜ ∝ n⁻¹’² ∝ P¹’²

At some point σₜ will be reduced down to level of systematic errors imposed by telescope, atmosphere, host star stability, or data reduction technique. Now, σₜ is fixed so we can combine proportionalities.
σₜ(min) ∝ Mₚ⁻¹’³ P¹’² = const
∴ Mₚ¹’³ ∝ Rₚ ∝ P¹’²
⇒ Mₚ ∝ P³’² (at fixed σₜ)

Question 53

Limitations for TTV

Answer 53

Transit photometry can be affected by intrinsic variability of the host star:
* long-term variability due to changing observing conditions or stellar variability. Can be “de-trended” from data by fitting for a general slope or curvature in the baseline behaviour.
* short-timescale periodic variability e.g. due to star spots. Can often be modelled as it may vary on timescales related to the
stellar rotation rate but unrelated to the planet period.
* short-term stochastic variability due to helioseismology. Sets a
limit on the precision of transit detection. Surveys have tended to pick “quieter” F, G and K-type stars to minimize this

Question 54

Pros/cons/limitations for microlensing

Answer 54

The lightcurve is time symmertric, with a simple, smooth functional form.

It is achromatic, wavelength independent as it is purely a geometrical effect.

It is non-periodic. Low probability effect and so shouldn’t repeat for the same source star.

Stars in our Galaxy have motions with respect to each other. This means that microlensing is a transient effect lasting only whilst
the lens is closely aligned to the source on the sky.

Question 55

Multiple lens system

Answer 55

If a planet orbits a host star both the planet and its host may act as a lens. They would constitute a binary lens system, which can produce more complex lightcurves than a single lens.
Adding lenses increases complexity of the microlensing effect:
* More (unresolvable) images are produced. For N-lens system, the maximum number of images produced is N²+1. 2-lens (binary lens) system, up to 5 images may be produced.
* Lensing geometry no longer circularly symmetric, instead exhibits a form of astigmatism where background star light is focused along locii. Projections of
these locii back onto the source plane are caustics.
* When a point source lies at the position of a caustic, the magnification diverges. See Fig(7.33 and 7.34) for examples.

Caustics:
The central caustic is always located close to the host star.
Very large body separation looks like 2 single lenses (point caustics).
Very small body separation looks like a single lens with double the mass.

Almost no evidence of planet if no caustics crossed.
Planet signal at central peak if central caustic crossed.
Strong peak away from central peak if planetarhy caustic crossed.

Question 56

Microlensing for FFPs

Answer 56

Alot of microlensing events have been oberved with a potentially large population of Earth/Superearth mass FFPs (Free-floating planets)

Very short duration implies their size.

Question 57

Dificulties for direct imaging

Answer 57

Extremely difficult due to
1) the small angular separation of planets from their host
2) the brightness contrast between the host star and its planets

Issue (1) can be addressed using adaptive optics, which enables ground-based telescopes to achieve close to space-based resolution. [alternative is make observations from space, but this is relatively expensive.]

Issue (2) can be addressed by the use of coronagraphy, where the central light of the host star is minimized in order to enhance the flux contrast of the planet.

Question 58

Airy disk

Answer 58

Airy disk represents the ideal image of a star that would be formed by a diffraction-limited telescope with a clear circular aperture.
Separation between the central maximum and first minimum
defines the Rayleigh diffraction limit at wavelength λ:
θ ≃ 1.22 λ/D
for a telescope of aperture D.

In practise, exoplanets must be located beyond 2λ/D from their host in order to have a chance of being detected.

Question 59

Point spread function

Answer 59

Point spread function (PSF) describes how the light from a point source is imaged through an optical system. It takes account of all factors within the optical system which may modify the resulting image. These will include:
* Diffraction
* Wavefront distortion due to turbulent motions in Earth’s atmosphere (seeing)
* Telescope movement relative to source whilst image is being captured (jitter)
* Misalignments between components within the optical assembly (collimation errors)
* Focussing errors
* Internal scattering of light off reflective surfaces within optical assembly.
* Resolution of the detector (eg the pixel size of a CCD array).

Seeing or detector resolution are often the limiting factors for ground-based image resolution

Question 60

Strehl ratio

Answer 60

Ratio of the PSF peak intensity compared to the central intensity of the Airy disk computed for the
same optical system.

Measures how close PSF of an optical system comes to the limiting case of an Airy disk.

An optical system with a high Strehl ratio is better able to detect closer exoplanets closer around a host star.

Question 61

Adaptive optics and Sheck-Hartmann sensor

Answer 61

Adaptive optics (AO) is used by ground-based telescopes to counteract seeing and to try to recover PSF profiles closer to the diffraction limit.
AO systems comprise deformable optics, such as a thin primary mirror whose shape is actively maintained by hundreds of actuators under its surface. AO systems correct the shape of the primary (or secondary mirror) in response to measurements of the shape of the wavefront from bright guide stars. Since atmospheric turbulence can
affect image quality on millisecond timescales, wavefront
measurements must be at frequencies in the kHz range. One of the most common types of wavefront sensor is the Shack-Hartmann sensor.
Sensor uses a lenslet array to view how guide stars are displaced from a distorted wavefront. This information is used to correct an adaptive mirror to correct the wavefront.

Question 62

Coronagraphy

Answer 62

A coronagraph reduces light from a star in order to increase the brightness contrast of nearby fainter objects.
Near diffraction-limited images typically exhibit central core with concentric rings (Airy disk).
The common Lyot coronagraph design reduces effect of the core using a central occulting mask, whilst another mask (a stop) reduces the effects of the rings.
Masks can either occult the light or split and phase-shift it so that it is minimized though self-interference.

Question 63

Qualitative dependence of RV
(inc. planet mass vs host separation and planet mass vs period.)

Answer 63

• Amplitude of RV signal scales with Mₚsin(i) in limit Mₚ ⟨⟨ M⋆
• Signal stronger for low mass hosts and short period (smaller host separation a) planets
• Fixed host mass M⋆ at limiting sensitivity K ~ σᵣᵥ = const, see that Mₚsin(i) ∝ P¹’³
• Full period should be sampled within survey lifetime
• Signal depends on sensitivity, stability and spectral resolution of the spectrograph

Log P vs log Mₚsin(i)
linear K ~ σᵣᵥ, above detectable. Undetectable for P ⟩ Tₛᵤᵣᵥₑᵧ

Question 64

Qualitative dependence of
astrometry on a graph of planet mass vs host separation and planet mass vs period.

Answer 64

Log P vs log Mₚ
-ve linear θ ~ σ꜀, above detectable. P ⟩ Pₘₐₓ undetectable as survey time not long enough.

Fig(4.6)

Question 65

Dependencies for transit method (inc. a graph of planet mass vs host separation and planet mass vs period)

Answer 65

• Larger planet radius, deeper lightcurve.
• Radius is correlated with mass Mₚ. For rocky (lower mass) planets, may expect a roughly characteristic planet density (for solar system rocky planets this is 4-5 g/cm³), in which case Mₚ ∝ Rₚ³
• Grazing transit curved shallower lightcurve, only becomes straight at/near full transit

Log P vs log Mₚ
linear Mₚ ∝ P³’², above detectable. Undetectable for P ⟩ Tₛᵤᵣᵥₑᵧ

Question 66

Qualitative dependence of microlensing on a graph of planet mass vs host separation and planet mass vs period.

Answer 66

Microlensing is capable of detecting low-mass planets. But its sensitivity declines for planet orbits much larger or smaller that the Einstein radius of the host star.
Planets at very large orbital separation, including unbound (free-floating) planets can be detected as short timescale single-lens events.

Log(a) vs log(Mₚ)
almost parabolic, detectable bound planets above curve, to the right planets unbound.
Fig(7.49)

Question 67

Qualitative dependence of direct imaging on a graph of planet mass vs host separation and planet mass vs period.

Answer 67

Log(a) vs log(Mp)
Mp too low, planet too faint to image. a too low, planet too close to host. Rectangle in top right of detectable planets.

Question 68

Host separation vs mass by technique

Answer 68

Fig(9.3, 9.5)

Question 69

Exoplanet occurrence rate and its dependencies

Answer 69

Frequency of exoplanets per host. In principle, depends on a number of parameters, including:
* the planet mass or radius
* orbital separation
* orbital eccentricity
* host star type, metallicity, age, etc

Some ‘facts’:
* The occurrence rate integrated over all period and mass ranges suggests that almost all stars likely host planets.
* Planets with masses from ~3-10 Earth mass (“Super-Earths”) are among the most common planets, despite being absent in our own Solar System.
* Giant planets appear to be less common around older stars with lower metallicity. Super-Earths show no such dependence.
* Comparison of RV and microlensing statistics suggest that low mass (M dwarf) stars have fewer giant planets but more Super-Earth sized planets.
* Where Super-Earths are found, they tend to be part of a multiple planet system as opposed to a 1-planet system

Question 70

Planet correlations

Answer 70

No strong evidence for wide-spread orbit resonance effects, but where there are indications of orbit resonance it tends to involve higher mass planets.

Probabilities of hot, cold jupiters and super earths around sun-like stars:
* P(HJ) ≈ 1%
* P(CJ) ≈ 10%
* P(SE) ≈ 30%
* P(CJ|HJ) ≈ 70%
* P(HJ|CJ) ≈ 7%
* P(CJ|SE) ≈ 30%
* P(SE|CJ) ≈ 90%
Jupiter’s between M(saturn) and M(jupiter)
Hot ⟨ 0.1AU, cold between 1 and 10 AU
SE mass between earth and neptune, between 0.1AU and 1AU

Question 71

Fulton Gap

Answer 71

Gap in the planet size distribution for hot exoplanets at around 1.5-2 Earth radii.
Gap occurs at larger radii for close-in planets around more massive stars.
Possible evidence that origin is photoevaporation of low density atmospheres.
Larger planets may be Neptune-like, smaller planets more rocky?

Question 72

RV host star biases

Answer 72

RV method requires high velocity precision to find low mass planets or planets at large host separation. This in turn requires “quiet” hosts with low surface activity. Such activity gives rise to RV jitter, sources include:
* Pressure (p-mode) fluctuations within host interior, timescale of minutes.
* Granulation, with timescales of hours.
* Star spots and plages, on timescales of days to months
* Cycles of magnetic activity, on timescales of years

Question 73

Transit host star biases

Answer 73

• The sources of variability which give rise to RV jitter also typically give rise to photometric variability in the host flux, so cause fluctuations in transit lightcurves on similar timescales.
• Longer term host variability causes a trend in the lightcurve which can be easily removed.
• The nature of shorter timescale
  variability may be identifiable through multi-wavelength observations but is more difficult to characterize and remove.

Question 74

Microlensing host star biases

Answer 74

• Microlensing signals are typically large and short lived and so are not usually affected by host variability.
• Host stars of microlensing planets are usually low mass M or K dwarf stars, as these are the
  most numerous type of stars in our Galaxy.
• Microlensing planet samples therefore do not sample very well the frequency of planets around solar-type (G type) stars, or around heavier stars.
• Most of the microlenses are likely to reside in the bulge (though often we do not know their exact location) and so microlensing does not probe planets around disk stars (like the Sun) very efficiently.

Question 75

Circumbinary planets

Answer 75

• Handful of systems known to date
• Orbits typically just outside of the instability zone and therefore appear to be dynamically stable (just!)
• Numerical simulations so far indicate it is very difficult to form stable circum-binary planetary systems with planets at the locations they have been discovered. But it is possible to form planets which start out further out but then migrate inwards to their current location.
   Evidence for the migration of planets

Question 76

Approx. planetary mass-radius relation

Answer 76

Mₚ
= (Rₚ/R🜨)^3 M🜨 for (Rₚ ⟨ R🜨)
= (Rₚ/R🜨)^2.06 M🜨 for (R🜨 ⟨ Rₚ ⟨ Rⱼᵤₚ)
Above Mⱼᵤₚ, planets expected to have constant radius. Also observed for lowest mass stars and brown dwarfs.

In many cases, eavier planets are larger than predicted, these planets are close to their hosts and are believed to have bloated sizes due to stellar radiation.

Question 77

How do we measure the density of exoplanets and how can this constrain the composition of a
planet?

Answer 77

Principal method is to obtain both transit and RV measurements. RV measurement gives Mp sin(i) and transit measurement gives Rp/R⋆ and also knowledge that sin(i) ≈ 1.
R⋆ is generally already known or obtainable from spectroscopy
follow-up.

For multi-planet systems it is also possible to obtain an exoplanet mass limit from TTV measurements rather than through RV, allowing a density limit purely from transit observations.

Planet composition can be constrained if the planet mass and radius is known as these provide the average density.

“Idealized” planet models are used to represent gas giants, ice giants and rocky planets:
* H/He planet to represent gas giants
* H/He + ices for ice giants
* “Water World” of pure H₂O as an extreme case for modeling
possible oceans.
* MgSiO₃ to represent a rocky planet
* Iron planet as an extreme case to model planets with large
cores

These can be plotted as curves on a radius vs mass diagram. For fixed composition the Mp – Rp relation will be controlled by the equation of state.

Question 78

What are the primary compositional differences between rocky planets, ice giant planets and gas giant planets?

Answer 78

Gas giants
* Predominately H+He composition reflecting proto-stellar gas abundance.
* Planets above 0.1 MJup are broadly consistent with expectations of this composition.
* Many have lower density than standard H+He mix, they are associated with high Teq, likely stellar irradiation bloating.
* A few have higher densities, perhaps indicating a substantial Fe core. (Jupiter is thought to have a core of around 4-10 M⊕). Host star metallicity may be a determinant of size of giant planet core.

Ice giants
* mostly comprise a mantle of water, methane, ammonia ices, but with an H+He atmosphere.

Rocky planets
* Earth composition dominated by an Fe core and a mantle comprising mostly MgO and SiO₂.
* Density of other inner planets suggests similar composition but with differing mass ratios of core:mantle, and no substantial water.
* Earth-like planets are often characterised by a 3-component model of an Fe core, MgSiO₃ mantle+crust and H₂0 surface
* Planets below 0.1 MJup not as well studied due to low RV amplitude and small transit depths.
* Density versus mass shows evidence of a minimum density for planets of ~ 0.2 g cm-3
* Current results consistent with a broad spectrum of compositions, from rock/iron planets to largely gaseous.
* Modelling multi-component internal composition suffers from degeneracies if only the mass and radius are known.

Question 79

What techniques are used to measure the chemical composition in the atmosphere of planets?

Answer 79

Emission spectroscopy
Obtaining spectrum of intrinsic planet thermal emission. This may be obtained through direct imaging or as excess emission from host flux observed as a planet passes out from secondary transit (when host eclipses planet).

Reflection spectroscopy
Obtaining spectrum of host light which has been reflected from atmosphere of an orbiting planet. At short (optical) wavelengths reflected spectrum will typically dominate over intrinsic planet emission spectrum.

Transmission spectroscopy
Obtaining a spectrum of host light which shines through the planetary atmosphere during primary transit. The net flux which passes through will depend upon the atmospheric optical depth at the observation wavelength. Observing over a range of wavelengths allows the construction of an absorption spectrum of the planet atmosphere.

Question 80

Transmission spectroscopy for atmospheric composition

Answer 80

• During primary transit the planet atmosphere absorbs host
  star light.
• During secondary transit the planet is hidden behind the host. Only the host star spectrum is observed.
• Difference can be taken to isolate planet spectrum.
• Generally, the atmospheric opacity is a function of wavelength. If an exoplanet atmosphere is optically thick at an observation wavelength λ1 then the planet apparent transit disk with be larger by an amount given by the column height of the atmosphere.
• At another wavelength λ2 the same atmosphere may be largely transparent, in which case the transit disk will reflect the solid surface of the planet (in the case of a rocky planet).
• The depth of the transit and the variation of the transit ingress/egress gradient with wavelength therefore provides a measure of the atmosphere absorption characteristics.
• Increasing opacity, wider and deeper lightcurve.

Question 81

Key stages of planet formation and their associated timescale

Answer 81

Cloud collapse
* Dense cool molecular clouds collapse to form protostars with gaseous + dusty disks (protostellar disk).
* Conservation of angular momentum: collapse naturally leads to the formation of a disk.
* Collapse timescale of initial cloud is short (free-fall timescale of the gas cloud): few x 10⁵ yrs.

Massive disk phase
* For matter to accrete onto protostar from the disk requires angular momentum to be removed from disk (transported outwards). Processes which can do this are typically inefficient so disk mass builds up and can become substantial.
* Eventually becomes unstable to disk perturbations - rapid accretion onto the protostar (disk mass loss).
* Stability of the disk is given by the Toomre Q parameter:
Q = cₛΩ/πGΣ
where cₛ is the gas sound speed, Ω is the gas angular velocity, and Σ is the disk surface density.
* Disk unstable if Q < 1 (too dense or not rotating sufficiently to provide support against gravity).
* Massive disks evolve to low mass (Toomre-stable) disks on a
timescale of a few x 10⁶ yrs.
* Note that massive stars can support more massive stable disks.

Low mass disk phase
* Protostar attains significant fraction of its final mass through accretion of disk gas.
* Disk evolves to a relatively low-mass configuration - protoplanetary disk (rather than
protostellar).
* During the massive disk phase the disk luminosity was characterised by conversion of
gravitational energy of accreted gas. Now during low-mass disk phase, disk luminosity is due primarily to re-processed star light.
* Size of planets that can form depends on overall mass of disk and rate at which the gas is accreted or dispersed.
* After typically ~ 10 million years, most of the gas has gone so planets can only grow through hierarchical growth.

Debris disk
* MRI produces self-sustaining turbulence which can act to efficiently transport angular momentum out of the disk.
* Dust can coagulate to form planetesimals of 50-100 m size which can then hierarchically combine through gravitational encounters.
* Debris disks around some nearby stars have been directly imaged due to their infrared excess.
* At this point planets can grow only through hierarchical growth of planetesimals via collisions.
* Collisional energy determines whether a collision results in a merger or destruction of the planetesimals.
* The physics is complex: requires detailed simulation work.

Question 82

Show analytically why it is hard to form planets out of a differentially rotating gas disk without supplying some source of turbulence, such as a magnetic field?

Answer 82

See (11.7 to 11.13)

Rayleigh stability criterion
dL^2/da ⟩ 0
This ensures ω^2 ⟩ 0 so disk perturbations cannot grow.

Question 83

Do you understand MRI?

Answer 83

Magneto-rotational instability

Outer surface of the gas disk is expected to be ionised by cosmic rays and proto-stellar radiation. The presence of even a weak
magnetic field can have a dramatic effect on angular momentum transport.

Question 84

Show analytically why MRI can help to form planets

Answer 84

Weak magnetic field perpendicular to disk plane provides spring like restoring force with spring const.
K ∝ kᶻ²B²
where B is mag field strength and kᶻ is vertical wavenumber

see (11.14 to 11.17)
-ve solutions from weak field allow instability

Question 85

Core accretion

Answer 85

• Idea that massive planets form first through the formation of rocky planetary cores. Once these cores exceed around 10M⊕ they become sufficiently massive to invoke rapid runaway accretion of the surrounding
  gas.
• Forming massive cores is in the inner regions of protoplanetary disks is problematic as abundance of metals (ie dust grains) is relatively low.
• Further out in the disk, volatiles are cool enough to form solid ices: can combine with dust grains to form larger cores.
• The distance beyond which ices are able to form is known as the snow line and depends on host
  luminosity.
• Core accretion could be a primary mechanism for forming ice and gas giant planets in the outer disk, whilst preferentially forming only smaller rocky planets closer in.
• Our own Solar system follows this architecture but, as we have seen (eg hot Jupiters), many systems apparently do not.

Question 86

Snow line

Answer 86

The snow line refers to the distance from a forming star (or the solar nebula) in which it is cold enough to form volatile compounds, including water, methane, CO2, CO and ammonia, among others.

Each substance has its own snow line. Can be used to separate terrestrial planets from gas giants.

The radial position of the snow line can also evolve with time, as the nebula evolves and the star forms.

The snow line also sometimes is a short-hand term for the “water snow line” – the location where water is in a stable ice-state even under direct star light.

The current snow line for water in the solar system is 5 AU.

Question 87

Explain planet migration

Answer 87

Type I migration:
Small planet embryos (of order Earth mass) may produce spiral density perturbations in the gas disk. The outer wave produces a torque which reduces the planet orbital angular momentum, causing it to migrate inwards.

Type I migration is thought to give rise to the “planet desert” of intermediate mass planets at smaller host separation.

Type II migration:
The planet is more massive and has accreted enough gas to clear its neighbourhood (produces a gap in the gas disk). But gas continues to accrete inwards towards the host so the gap also migrates inwards on an accretion timescale. The outer disk gas falls onto the planet, causing viscous drag which produces orbital angular momentum loss. The planet also migrates inwards.

Type II migration is thought to give rise to the population of hot Jupiters detected by transit and RV surveys.

Question 88

How can planet migration help us understand some of the key features seen in planet formation simulations?

Answer 88

• Hot Jupiters arise from the migration of gas giants formed further out beyond the ice line
• Close in planets with masses of around 10-100 M⊕ should be comparatively rare (“planet desert”), a result of Type I migration.
• Whist RV and transit methods are sensitive to hot exoplanets, we need to take into account migration effects to link these observations with planet formation models.
• Low mass planets beyond the ice line should today be located more or less where they formed. They are a good diagnostic of “in-situ” planet formation free of assumptions of subsequent migration effects.
• The microlensing technique typically also probes the low-mass planet regime beyond the ice line. It is therefore an ideal technique to probe in-situ planet formation.

Question 89

Habitable zone

Answer 89

Conditions necessary to sustain
life.

“Requirements”:
* A terrestrial planet of similar size to Earth (solid surface)
* A planetary atmosphere comprising N2, H2O and CO2 (main constituents controlling climate)
* A host star which is stable over an evolutionary timescale.
* The ability of the planet to host and retain water over an evolutionary timescale

The requirement of hosting water leads to an inner and outer edge to the HZ.

Continuously Habitable Zone (CHZ) is used to refer to HZ boundaries which always remain within habitable limits on Seff (effective flux, outgoing planet IR / incoming stellar flux) throughout the main sequence life of a star

Question 90

η⊕ (Eta Earth)

Answer 90

η⊕ refers to the average number Earth-like planets per star which are located within the HZ of their host.
Since M dwarf stars are the most common stellar type the abundance of habitable planets may well be determined by their frequency around M dwarfs.
A recent study of Kepler planet candidates found around M dwarf stars finds that η⊕ ≈ 0.5 for planets with radii within a factor 2 of Earth.
RV surveys have found a similarly high abundance.
Potentially habitable planets appear to be a common occurrence around the most common types of stars in our Galaxy.