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International
Astronomy and Astrophysics
Competition (IAAC) 2021
Solutions for the Pre-Final Round
A.
Problem A.1. Equatorial Coordinate System (4 Points)
Astronomers need to identify the position of objects in the sky with very high precision. For that, it is essential
to have coordinate systems that specify the position of an object at a given time.
One of them is the equatorial coordinate system that is widely used in astronomy.
Explain how the equatorial coordinate system works.
equatorial coordinate system
right ascension α declination δ
α 000 235959
δ 0 00
What is the meaning of that often occurs together with equatorial coordinates?
≈
The object NGC 4440 is a galaxy located in the Virgo Cluster at the following equatorial coordinates
(J2000): 12h27m53.6s(right ascension), 000 (declination). The Calar Alto Observatory is located in
Spain at the geographical coordinates . N and . W.
Is the NGC 4440 galaxy observable from the Calar Alto Observatory?
h
altitude horizontal coordinate system
geocentric
=ϕ−δ+ε+h,
ϕ
ε
≈
≈. ≈.×
ε→0
=⇒h ≈ −ϕ+δ= −. +12 + 17
60 +36
3600=. ≈000 ≥.
is
http://www.caha.es/about- caha-mainmenu- 93/introduction-mainmenu-49
)e
live
ot
siah
eqaler
S
bene
al/celeshl
ocal
hon
2or
δ ϕ h
Problem A.2. Resolution of Telescopes (4 Points)
Telescopes are an essential tool for astronomers to study the universe. You plan to build your own telescope
that can resolve the Great Red Spot on the surface of Jupiter at a wavelength of [λ:=] . The farthest
distance between the Earth and Jupiter is [∆:=] × and the Great Red Spot has currently a diameter
of [d:=] .
Use the criterion to determine the diameter of the lens’ aperture of your telescope that is needed
to resolve the Great Red Spot on Jupiter.
θ= 1.22 λ
D,
θ λ D
D= 1.22 λ
θ.
θ
R
θ
2≈ θ
2≈R
∆=d
2∆,
d ∆
θ≈d
∆,
D≈1.22 λ∆
d= 1.22 ·×−·. ×
. ×≈1·6·10−7+12
3
2×107= 6 ·2
3·10−7+12−7= .
https://en.wikipedia.org/wiki/Angular_resolution#The_Rayleigh_criterion
θ≈θ
θ=
∞
X
n=0
1
n!n( θ)
θn(0) θn=(0)
| {z }
0
+1
2(0)
| {z }
1
θ+Oθ2=θ+Oθ2.
Impacts have formed many craters on the Moon’s surface. You would like to study some of the craters with
your new telescope. The distance between Moon and Earth is [∆:=] .
What is the smallest possible size of the craters that your telescope can resolve?
λ=
λ=
≈1.22 λ∆
D= 1.22 ·.×−·. ×
×−≈1·6·4
4×10−7+8+2 = .
Problem A.3. Total Solar Eclipse (4 Points)
A total solar eclipse occurs when the Moon moves between the Earth and the Sun and completely blocks out the
Sun. This phenomenon is very spectacular and attracts people from all cultures. However, total solar eclipses
can also take place on other planets of the Solar System.
Determine for each of the following moons if they can create a total solar eclipse on their planet.
[…]
Note: The radius of the Sun is [R:=] ×.
θ
θ
2r δ
2R ∆
θ:=2r
δ≥2R
∆=:θ
⇔θ
θ=2r
δ
∆
2R=r
δ
∆
R≥1.
r δ ∆ θ/θ
11 9,376 2.28 ·1080.38
2,410 1.883 ·106 7.79 ·1081.43
2,574 1.222 ·106 1.433 ·1094.34
761 5.84 ·105 2.875 ·1095.38
Problem A.4. Special Relativity – Part I (4 Points)
Special relativity has become a fundamental theory in the 20th century and is crucial for explaining many
astrophysical phenomena. A central aspect of special relativity is the transformation from one reference frame
to another. The following transformation matrix gives the transformation from a frame at rest to a
moving frame with velocity valong the z-axis:
γ0 0 γ β
0 1 0 0
0 0 1 0
γ β 0 0 γ
β=v
γ
γ=1
p1−β2.
State and explain the two traditional postulates from which special relativity originates.
V
Draw a plot of the Lorentz factor for 0≤β≤0.9to see how its value changes.
https://einsteinpapers.press.princeton.edu/vol2-trans/157
One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts in a spaceship
that is passing by the Earth with a high velocity.
Are clocks ticking slower for the people on Earth or for the astronauts on the spaceship?
their own clocks
their own clocks
dierent compared
slower than their own clocks
slower than their own clocks
t
x
y
z
,
t, x, y, z t0, x0, y0, z0
z
t0
x0
y0
z0
=
γ0 0 γ β
0 1 0 0
0 0 1 0
γ β 0 0 γ
t
x
y
z
.
t0=γt+γ β z =γ(t+β z) = γt+v
z.
e1:= (t1, x1, y1, z1)7→ (t1
0, x1
0, y1
0, z1
0) e2:= (t2, x2, y2, z2)7→ (t2
0, x2
0, y2
0, z2
0)
∆t0:=t2
0−t1
0=γ(t2+β z2)−γ(t1+β z1)
=γ((t2−t1) + β(z2−z1)) =:γ(∆t+β∆z)
z0=γ β t+γ z =γ(βt+z) = γ(v t +z).
∆z0:=z2
0−z1
0=γ(v(t2−t1)+(z2−z1)) =:γ(v∆t+ ∆z)
⇔∆z=∆z0
γ−v∆t.
∆t0=γ(∆t+β∆z)
⇔∆t=∆t0
γ−β∆z
=∆t0
γ−β∆z0
γ−v∆t
=∆t0
γ−β∆z0
γ+β v ∆t
⇔∆t=∆t0
γ−β∆z0
γ+v2
∆t
⇔∆t=∆t0
γ−β
∆z0
γ+v2
2∆t
⇔1−v2
2∆t=1
γ2∆t=∆t0
γ−β
∆z0
γ
⇔∆t=γ∆t0−β
∆z0.
v= 0 =⇒β= 0 =⇒γ= 1 =⇒∆t0= ∆t.
∆z0= 0 =⇒∆t=γ∆t0=⇒∆t0=∆t
γ.
v6= 0 =⇒γ > 1 =⇒∆t0<∆t.
How fast must the spaceship travel such that the clocks go twice as slow?
γ=1
p1−β2
⇔γ2=1
1−β2
⇔1−β2=1
γ2
⇔β2= 1 −1
γ2
⇔β=v
=r1−1
γ2
⇔v=r1−1
γ2.
∆t0=∆t
2=⇒γ= 2 =⇒v=r1−1
22=r3
4=√3
2≈. .
B.
Problem B.1. Space Cannon (6 Points)
Scientists are developing a new space cannon to shoot objects from the surface of the Earth directly into a low
orbit around the Earth. For testing purposes, a projectile is red with an initial velocity of .
vertically into
the sky.
Calculate the height that the projectile reaches, …
assuming a constant gravitational deceleration of .
considering the change of the gravitational force with height.
Note: Neglect the air resistance for this problem. Use . ×−
for the gravitational constant,
for the Earth’s radius, and . × for the Earth’s mass.
h
T(v(0)) + U(0)
|{z}
0
=T(v(t(h)))
| {z }
0
+U(h)
⇔U(h) = T(v(0))
⇔
h=1
2
v(0)2
⇔h=v(0)2
2=
2
2·.
≈ .
¨r=−
r2
⇔¨r=−
r2
⇔¨r˙r=− ˙r
r2
⇔1
2
t˙r2=
t1
r
⇔1
2˙r2=
r
⇔1
2˙r2=
r
Problem B.2. Shock Wave (6 Points)
This year’s qualication round featured a spaceship escaping from a shock wave (Problem B). The crew survived
and wants to study the shock wave in more detail. It can be assumed that the shock wave travels through a
stationary ow of an ideal polytropic gas which is adiabatic on both sides of the shock. Properties in front and
behind a shock are related through the three – jump conditions (mass, momentum, energy
conservation):
ρ1v1=ρ2v2
ρ1v12+p1=ρ2v22+p2
v12
2+h1=v22
2+h2
where ρ,v,p, and hare the density, shock velocity, pressure, and specic enthalpy in front (1) and behind (2)
the shock respectively.
Explain briey the following terms used in the text above:
stationary ow
polytropic gas
p V n=
p V n polytropic index
specic enthalpy
h:=H
,
H
H:=U+p V,
U p V
Show with the – conditions that the change in specic enthalpy is given by:
∆h=p2−p1
2·1
ρ1
+1
ρ2.
ρ12v12=ρ22v22
⇔ρ22v22−ρ12v12= 0.
v12=ρ2v22+p2−p1
ρ1
, v22=ρ1v12+p1−p2
ρ2
.
∆h:=h2−h1=v12
2−v22
2
=1
2v12−v22
=1
2ρ2v22+p2−p1
ρ1−ρ1v12+p1−p2
ρ2
=1
2ρ2v22
ρ1
+p2−p1
ρ1−ρ1v12
ρ2
+p1−p2
ρ2
=1
2ρ2v22
ρ1
+p2−p1
ρ1−ρ1v12
ρ2−p1−p2
ρ2
=1
2ρ2v22
ρ1
+p2−p1
ρ1−ρ1v12
ρ2
+p2−p1
ρ2
=p2−p1
2ρ2v22
ρ1(p2−p1)+1
ρ1−ρ1v12
ρ2(p2−p1)+1
ρ2
=p2−p1
2ρ2v22
ρ1(p2−p1)−ρ1v12
ρ2(p2−p1)+1
ρ1
+1
ρ2
=p2−p1
2
: 0
ρ22v22−ρ12v12
ρ1ρ2(p2−p1)+1
ρ1
+1
ρ2
=p2−p1
21
ρ1
+1
ρ2.
The general form of ’s law is fullled on both sides of the shock separately:
v2
2+Φ+h=,
where Φis the gravitational potential and a constant.
Assuming that the gravitational potential is the same on both sides, determine how the constant changes
at the shock front.
v12
2+Φ+h1=1,v22
2+Φ+h2=2
=⇒∆:=2−1=v22
2+Φ+h2−v12
2+Φ+h1=v22
2+h2−v12
2+h1
= 0.
Explain whether ’s law can be applied across shock fronts.
Φ
∆=2−1= 0 =⇒1=2=:
=⇒v12
2+Φ+h1==v22
2+Φ+h2,
⇔v12
2+h1=v22
2+h2,
∆6= 0 =⇒16=2
Problem B.3. Interplanetary Journey (6 Points)
A space probe is about to launch with the objective to explore the planets Mars and Jupiter. To use the lowest
amount of energy, the rocket starts from the Earth’s orbit (A) and ies in an elliptical orbit to Mars (B), such
that the ellipse has its perihelion at Earth’s orbit and its aphelion at Mars’ orbit. The space probe explores
Mars for some time until Mars has completed 1/4 of its orbit (C). After that, the space probe uses the same
ellipse to get from Mars (C) to Jupiter (D). There the mission is completed, and the space probe will stay around
Jupiter.
The drawing below shows the trajectory of the space probe (not drawn to scale): […]
Below you nd the [orbital] period and the semi-major axis of the three planets:
How many years after its launch from the Earth (A) will the space probe arrive at Jupiter (D)?
∆t→=T→
2
T→2
T2=a→3
a3.
=⇒T→=sa→3T2
a3=Tsa→3
a3≈sa→3
( )3· =v
u
u
ta+a
23
( )3=v
u
u
t. +.
23
( )3
=s(2.52)3
8≈r16
8=√2,
:=. ≈
=⇒∆t→=√2
2≈.
∆t→=T
4=1
4·
≈.
T→=sa→3
( )3=v
u
u
ta+a
23
( )3=v
u
u
t. +.
23
( )3
=s(6.72)3
8≈r303.46
8≈.
∆t→=T→
2≈. .
∆t= ∆t→+ ∆t→+ ∆t→≈. +. +. =. .
Problem B.4.
Space and time are interconnected according to special relativity. Because of that, coordinates have four compo-
nents (three position coordinates x,y,z, one time coordinate t) and can be expressed as a vector with four rows
as such:
t
x
y
z
The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the deep space
outside of our Milky Way. The Milky Way has a very circular shape and can be expressed as all vectors of the
following form (for all 0≤ϕ < 2π):
t
0
ϕ
ϕ
How does the shape of the Milky Way look like for the astronauts in the fast-moving spaceship? To answer
this question, apply the transformation matrix (see A.4) on the circular shape to get the vectors
(t0, x0, y0, z0)of the shape from the perspective of the moving spaceship.
L:=
γ β 0 0 γ
0 1 0 0
0 0 1 0
γ β 0 0 γ
t0
x0
y0
z0
=L
t
x
y
z
!
=L
t
0
ϕ
ϕ
=
γ β t+γ ϕ
0
ϕ
γ β t+γ ϕ
=
γ β t+γ ϕ
0
ϕ
γ(v t + ϕ)
,0≤ϕ < 2π
=⇒y0= ϕ=y,
z0=γ(v t + ϕ)
⇔z= ϕ=z0
γ−v t.
t= 0 =⇒z=1
γz0=p1−β2z0< z.
not
https://en.wikipedia.org/wiki/Terrell_rotation
Problem
B.4:
Special
Relativity -
Part
Il
(6
Points)
Space
and
time
are
interconnected
according
to
special
relativity.
Because
of
that,
coordinates
have
four
components
(three
position
coordinates
r,
J,
2,
one
time
coordinate
t)
and
can
be
ex
pressed
as
a
vector
with four rows
as
such:
ct
The
spaceship
from
problem
A.4
(Special Relativity -Part
1)
travels away from
the
Earth into
the
deep
space
outside
of
our
Milky
Way.
The
Milky
Way
has a very circular
shape
and
can
be
ex-
pressed as all vectors of
the
following form (for all 0 p <2m):
ct
sin
COS
p
a)
How
does
the
shape
of
the
Milky
Way
look like for
the
astronauts
in
the
fast-moving
space-
ship?
To
answer
this question, apply
the
Lorentz transformation matrix (see
A.4)
on
the
circular
shape
to get
the
vectors (ct',
r',
y,
2)
of
the shape from
the
perspective of
the
moving
spaceship.
(b) Draw
the
shape
of
the
Milky
Way
for a spaceship with a velocity of 20%, 50%,
and
90%
of
the
speed
of light
in
the
figure below (Note: The ring shape
fora
resting
spaceship
is
already drawn.):
1.86
0.75
9.59
9.25
+
0.00
K-O.Sc
-0.25
-0.50
VCSc
-0.1
1.00
-2.0 -1.5
1.6
.5
1.5
2.0
-2
z'
10
10
www.iaac.space
11/14
C.
Problem C.1. Earliest Galaxy Group (10 Points)
This problem requires you to read the following recently published scientic article:
https://iopscience.iop.org/article/10.3847/2041-8213/ab75ec
Answer the following questions related to this article:
What is the so called cosmic reionization process?
What are Lyαlines and why did the [researchers] want to observe them?
αα
n
n
α
What do the authors intend to point out with Figure 1 (see article)?
How is conrmed that the peaks seen in Figure 3 are actually from Lyαemissions?
α
Skewness
Sw ()
α
How are the bubble sizes of the galaxies estimated?
α
What is special about the ndings in the article and what are the scientic implications?
z= 7.7
References
The
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
Optics & Laser Technology