IPhO_2025_Q1-01
Hydrogen and galaxies
IPhO · 2025 · Fisika Modern
This problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. Throughout this problem we will only use , defined as
.
Part A - Introduction
Bohr model We assume that the hydrogen atom consists of a non - relativistic electron, with mass
, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit.
A. 1
Determine the electron's velocity in a circular orbit of radius .
0. 2pt
In the Bohr model, we assume the magnitude of the electron's angular momentum is quantized,
where
is an integer. We define
.
A. 2
Show that the radius of each orbit is given by
, where
is called the Bohr radius. Express
in terms of ,
, and and calculate its numerical value with 3 digits. Express
, the velocity on the orbit of radius
, in terms of and .
0. 5pt
A. 3
Determine the electron's mechanical energy
on an orbit of radius
in terms of ,
,
and . Determine
in the ground state in terms of ,
and . Compute its numerical value in eV.
0. 5pt
Hydrogen fine and hyperfine structures The rare spontaneous inversion of the electron's spin causes a photon to be emitted on average once per 10 million years per hydrogen atom. This emission serves as a hydrogen tracer in the universe and is thus fundamental in astrophysics. We will study the transition responsible for this emission in two steps. First, consider the interaction between the electron spin and the relative motion of the electron and the proton. Working in the electron's frame of reference, the proton orbits the electron at a distance
. This
produces a magnetic field
.
A. 4
Determine the magnitude
at the position of the electron in terms of
, , , and
.
0. 5pt
Second, the electron spin creates a magnetic moment
. Its magnitude is roughly
. The fine
structure is related to the energy difference
between an electron with a magnetic moment
parallel to
and that of an electron with
anti - parallel to
. Similarly, the hyperfine ( HF) structure is related to the energy difference
, due to the interaction between parallel and anti - parallel magnetic
F where is
moments of the electron and the proton. It is known to be approximately
where is
the proton mass.
Theory
English ( Official)
A. 5
Express
as a function of and
. Express the wavelength
of a photon emitted during a transition between the two states of the hyperfine structure and give its numerical value with two digits.
0. 5pt
Part B - Rotation curves of galaxies
Data
Kiloparsec: 1 kpc =
m Solar mass: 1 M
=
kg We consider a spherical galaxy centered around a fixed point . At any point , let
be the volumetric mass density and
the associated gravitational potential ( i. e. potential energy per
unit mass). Both and depend only on
. The motion of a mass located at , due to the field , is restricted to a plane containing .
B. 1
In the case of a circular orbit, determine the velocity
of an object on a circular
orbit passing through in terms of and
.
0. 2pt
Fig. 1 ( A) is a picture of the spiral galaxy NGC 6946 in the visible band ( from the 0. Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1 ( B) show experimental measurements of
for this galaxy. The central region (
kpc) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for
if the system were homogeneous in the bulge and keplerian (
with
) outside it, i. e. considering that the total mass of the galaxy is concentrated in the bulge.
B. 2
Deduce the mass
of the bulge of NGC 6946 from the red rotation curve in Fig. 1 ( B), in solar mass units.
0. 5pt
Comparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by
\rho_{m}(r) = \frac{C_{m}}{r_{m}^{2} + r^{2}} \tag{1}
where
and
are constants.
B. 3
Show that the velocity profile
, corresponding to the mass density in Eq. 1, can be written
. Express
and
in terms of
,
and .
0 1.8pt (Hints:
, and:
for
.)
Simplify
when
and when
. Show that if
, the mass
embedded in a sphere of radius with the mass density given by Eq. 1 simplifies and depends only on
and . Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1 ( A).
Part C - Mass distribution in our galaxy For a spiral galaxy, the model for Eq. 1 is modified and one usually considers the gravitational potential
where is the distance to the galactic plane ( defined by
), and
is now the axial radius and
a constant to be determined.
C. 1
Find the equation of motion on for the vertical motion of a point mass in such a potential, assuming is constant. Show that, if
, the galactic plane is a stable equilibrium state by giving the angular frequency
of small oscillations around it.
0. 5pt
From here on, we set
.
C. 2
Identify the regime, either
or
, in which the model of Eq. 1 recovers a potential of the form
with a suitable definition of
. Under this condition
no longer depends on . Express it in terms of
.
0. 6pt
Therefore, outside the bulge the velocity modulus
does not depend on the distance to the galactic center. We will use this fact, as astronomers do, to measure the galaxy's mass distribution from the inside. All galactic objects considered here for astronomical observations, such as stars or nebulae, are primarily composed of hydrogen. Outside the bulge, we assume that they rotate on circular orbits around the galactic center . is the sun's position and that of a given galactic object emitting in the hydrogen spectrum. In the galactic plane, we consider a line of sight
corresponding to the orientation of an observation, on the unit vector
(see Fig. 2).
C. 3
Determine
in terms of , ,
and
. Then, express in terms of
,
, and
.
0. 7pt
Using a radio telescope, we make observations in the plane of our galaxy toward a longitude
. The frequency band used contains the line, whose frequency is
GHz. The results are reported in Fig. 3.
C. 4
In our galaxy,
km s
. Determine the values of the relative radial velocity ( with 3 significant digits) and the distance from the galactic center ( with 2 significant digits) of the 3 sources observed in Fig. 3. Distances should be expressed as multiples of
.
0. 6pt
C. 5
On the top view of our galaxy ( in the answer box), indicate the positions of the sources observed in Fig. 3. What could be deduced from repeated measurements changing ?
0. 6pt
Part D - Tully - Fisher relation and MOND theory The flat external velocity curve of NGC 6946 in Fig. 1 is a common property of spiral galaxies, as can be seen in Fig. 4 ( left). Plotting the external constant velocity value
as a function of the measured total mass
of each galaxy gives an interesting correlation called the Tully -
Fisher
relation, see Fig. 4 ( right).
D. 1
Assuming that the radius of a galaxy doesn't depend on its mass, show that
the model of Eq. 1 ( part B) gives a relation of the form
where and should be specified. Compare this expression to the Tully -
Fisher
relation by computing
.
0. 4pt
D. 2
Using data for NGC 6946 in Fig. 1, estimate, within Newton's theory, the modulus of the acceleration
of a mass in the outer regions of NGC 6946.
0. 2pt
D. 3
Let be a mass on a circular orbit of radius with velocity
in the gravity field of a fixed mass . Within the MOND theory, with
, determine the Tully -
Fisher
exponent. Using data for NGC 6946 and / or Tully -
Fisher
law, calculate
to show that MOND operates in the correct regime.
0. 8pt
D. 4
Considering relevant cases, determine
for all values of in the MOND theory in the case of a gravitational field due to a homogeneously distributed mass with radius
.
0. 9pt
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