Long Question: T-13 Exomoon - · PDF filemoon barycentre • TDV Planet velocity around...
Transcript of Long Question: T-13 Exomoon - · PDF filemoon barycentre • TDV Planet velocity around...
Long Question: T-13
“Exomoon”
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Motivation • Exomoon
• Exomoon is a natural satellite of exoplanet
T-13, Exomoon
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Motivation • Kepler-1625 b I, an exomoon candidate,
orbits a Neptune-size gas giant roughly 4,000 light years from Earth.(Teachey et al. 2017)
• Several methods have been developed in order to detect exomoons, including:
• TTV Planet position around planet-moon barycentre
• TDV Planet velocity around planet-moon barycentre
T-13, Exomoon
[Credit: BBC]
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Motivation
• is moon phase • when the moon is in opposition to the star
T-13, Exomoon
fm0mf
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Motivation
T-13, Exomoon
sin( )2
m m p
TTV m
p p
a M Pf
a M
cos( )p m m
TDV m
m p p
P M af
P M a
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Motivation
• The relation between TTV signal ( )and TDV signal ( ) can be written as a linear function (Awiphan et al. 2013)
T-13, Exomoon
22
2 2 22 m m p
TDV TTV
m p p m
a M P
P a M P
2
TTV 2
TDV
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T-13, Exomoon
Rh = apM p
3M*
3
Motivation • Hill sphere
• A region around a planet within which the planet’s gravity dominates (Stable orbit).
• For massive host star, the radius of the Hill sphere of the system is approximately equal to the distance between planet and Lagrange point L1 or L2
A contour plot of the effective potential of a two-body system and Lagrange points [Credit: Wikipedia]
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T-13, Exomoon
Roche limit [Credit: Pearson Prentice Hall, Inc]
Rr = 2.44Rprp
rm3
Motivation • Roche limit
• The minimum orbital radius which a satellite can orbit without being torn apart by tidal force.
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T-13, Exomoon
Objectives
The ultimate goal is to find that the exomoon has
a stable orbit or not.
1. Solve equations of TTV and TDV signals2. Find exomoon parameters3. Find Hill radius and Roche limit radius of the system4. Does the exomoon have a stable orbit?
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T-13, Exomoon
Task 1 (7 marks)
• Show that the TTV signal can be written as,
• Projection distance of the planet to the planet-moon barycentre
sin( )2
m m p
TTV m
p p
a M Pf
a M
am =M p
Mm
apb aproj = apb sin( fm ) s TTV =aproj
ap
æ
èç
ö
ø÷
2p
Pp
æ
èç
ö
ø÷
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Task 2 (9 marks)
• Show that the TDV signal can be written as,
• Transverse velocity of planet around the planet-moon barycentre
T-13, Exomoon
cos( )p m m
TDV m
m p p
P M af
P M a
2pb pb
m
v aP
2cos( )m
trans m m
m p
Mv a f
P M
transTDV
p
v
v
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Task 3 (5 marks)
• Find mean transit duration
• Students need to determine• Distance of the planet-moon barycentre to the star• Mean transit duration
T-13, Exomoon
22 3
*
4
( )p
p
P aG M M
2P
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Task 4 (6 marks)
• Find moon period
• Linear function of TTV ( ) and TDV signal ( )
T-13, Exomoon
2
TTV 2
TDV
2
20.7432
mP
2 2 8 20.7432 1.933 10 daysTDV TTV
22
2 2 22 m m p
TDV TTV
m p p m
a M P
P a M P
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Task 5 (3 marks) • Find distance of the moon to the planet-moon barycentre• Kepler’s third law
Task 6 (4 marks) • Find moon mass• Linear function of TTV ( ) and TDV signal ( ) [Task 4]• Distance of the planet-moon barycentre to the star and transit
durataion [Task 3]• Distance of the moon to the planet-moon barycentre [Task 5]
T-13, Exomoon
22 34
m m
p
P aGM
2
2 8 21.933 10 daysm m p
p p m
a M P
a M P
2
TTV 2
TDV
2 2 8 20.7432 1.933 10 daysTDV TTV
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Task 7 (10 marks)
• Find radius of Hill sphere• Students need to determine
• Hill sphere function from Lagrange point
• Hill sphere radius
T-13, Exomoon
3
*
p
h p
MR a
xM 3
*3
p
m p
Ma a
M 3x
A contour plot of the effective potential of a two-body system and Lagrange points [Credit: Wikipedia]
52.2hR R
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Task 8 (3 marks)
• Find radius of Roche limit
• Roche limit radius
T-13, Exomoon
Rr = 2.44Rprp
rm3
12.0rR R
Roche limit [Credit: Pearson Prentice Hall, Inc]
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Task 9 (3 marks)
• Does the exomoon have a stable orbit?
• Students need to use• Distance between moon and planet-moon barycentre [Task 5]• Hill radius [Task 7]• Roche limit [Task 8]
Stable orbit
T-13, Exomoon
r m HR a R
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T-13, Exomoon
Knowledge
Basic Astrophysics/ Celestial Mechanics:• Newton’s laws of gravitation• Kepler’s law for circular and non-circular orbits• Roche limit• Barycentre• 2-body problem• Lagrange points
Stellar system/ Exoplanet:• Techniques used to detect exoplanet
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Modularity
T-13, Exomoon
Task 1• TTV Signal
Task 2• TDV Signal
Task 3•
• pa
Task 4• mP
Task 5•
ma
Task 6•
mM
Task 7• Hill sphere
Task 8• Roche limit
Task 9• Stable orbit