Exoplanets Flashcards
fig(week.slide#)
Definition of a planet.
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.
Definition of a dwarf planet
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
What is hydrostatic equilibrium?
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.
Kepler’s laws
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.
The definition and meaning of parameters which describe a planet’s orbit
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)
Barycentric view, star orbital parameters in terms of planetary ones
e⋆ = eₚ
a⋆ = aₚMₚ/M⋆
i⋆ = iₚ
Ω⋆ = Ωₚ
ω⋆ = ωₚ + π
ν⋆ = νₚ
Eccentricity equation
b² = (1-e²)a²
Position of a planet with respect to the system barycentre/focus
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
Derive the planet equilibrium temperature
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
Using a simple model atmosphere, explain why Earth’s temperature is higher than its equilibrium temperature
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)
Earth runaway greenhouse effect possible causes
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.
Photon noise limit and S/N ratio
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.
What is the radial velocity (RV) exoplanet detection technique?
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
Doppler formula
Relativistic:
λᵒᵇˢ = λₑₘ 1+(vᵣ/c)/√1−(vᵣ/c)²’
Non-relativistic:
Δλ/λ = vᵣ/c
What is the astrometry exoplanet detection technique?
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.
What is the transit exoplanet detection technique?
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.
What is the transit-timing variation (TTV) exoplanet detection technique?
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
Orbital resonance
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.
What is the microlensing exoplanet detection technique?
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.
What is the direct imaging exoplanet detection technique?
Viewing planet directly
Radial velocity calculations
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⋆
Astrometry angular displacement calculations
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
Astrometry centroid accuracy
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
Requirement to observe any transit
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.
