Long Question: T-13

“Exomoon”

Page 1

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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