IOAA 2017, Анализ на данни — Задача 2. The search for dark matter

• Part 2 Dark matter distribution, (D2.3) ρ₀ ≈ 1.42 × 10⁸ M☉/kpc³; r_C = 0.964 kpc (Method 2); r_C = 1.01 kpc (Method 1)

  Official solution; the less accurate estimate gives r_C ≈ 0.901 kpc.

• Part 2 Dark matter distribution, (D2.4) ρ₀ ≈ 1.43 × 10⁸ M☉/kpc³ (Solution 1, with r_C = 3.2 kpc)

  Handwritten result on the official graph D2.4: Solution 1.

(D2) The search for dark matter [75 marks]

A low surface brightness galaxy (LSB) is a diffuse galaxy with a surface brightness that, when viewed from the Earth, is at least one magnitude lower than the ambient night sky.

Some of its matter is in the form of “baryonic” matter such as neutral hydrogen gas and stars. However, most of its matter is in the form of invisible mass which is so called “dark matter”. In this question, we will investigate the mass of dark matter in a galaxy, the effect of dark matter on the rotation curves of the galaxy, and the distribution of dark matter in the galaxy.

The table below provides the data of a LSB galaxy named UGC4325. The galaxy is assumed to be edge-on. At every distance $r$ from the centre of the galaxy, we measure

1. $\lambda_{\mathrm{obs}}$, the observed wavelength of the H$\alpha$ emission line. The Hubble expansion of the Universe has already been excluded from the data.
2. $V_{\mathrm{gas}}$, the contribution of the gas component to the rotation due to $M_{\mathrm{gas}}$, derived from HI surface densities.
3. $V_*$, the contribution of the stellar component to the rotation due to $M_*$, derived from $R$-band photometry.

The rotational velocities of the test particle due to the gas component, $V_{\mathrm{gas}}$, and the star component, $V_*$, are defined as the velocities in the plane of the galaxy that would result from the corresponding components without any external influences. These velocities are calculated from the observed baryonic mass density distributions.

The mass of dark matter $M_{\mathrm{DM}}(r)$ within a volume of radius $r$ can be defined in terms of the rotational velocity due to dark matter $V_{\mathrm{DM}}$, the radius $r$ and gravitational constant $G$,

For the best estimate, the observed rotational velocity $V_{\mathrm{obs}}$ can be modeled as

The observed rotational velocity $V_{\mathrm{obs}}$ depends on the mass of the galaxy $M(r)$ within a volume of radius $r$ measured from the galaxy’s centre.

The mass density $\rho_{\mathrm{DM}}(r)$ of dark matter within a volume of radius $r$ can be modeled by a galaxy density profile,

where $\rho_0$ and $r_{\mathrm{C}}$ are the central density and the core radius of the galaxy, respectively. According to the density profile, the mass of dark matter $M_{\mathrm{DM}}(r)$ within a volume of a radius $r$ can be described by

Part 1 The mass of dark matter and rotation curves of the galaxy

(D2.1) In laboratories on Earth, H$\alpha$ has an emitted wavelength $\lambda_{\mathrm{emit}}$ of 656.281 nm. Compute the observed rotational velocities of the galaxy $V_{\mathrm{obs}}$ and the rotational velocities due to the dark matter $V_{\mathrm{DM}}$ at distance $r$ in units of km/s.

For the different values of $r$ given in the table, compute the dynamical mass $M(r)$ and the mass of dark matter $M_{\mathrm{DM}}(r)$ in solar masses. [21]

Solution:

Redshift: $z = \dfrac{\lambda_{\mathrm{obs}} - \lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}$ [0.5]

Observed rotation velocity: $V_{\mathrm{obs}} = zc$ [0.5]

Values correctly computed: 20.0 (0.5 for each values)

(D2.2) Create rotation curves of the galaxy on graph paper by plotting the points of $V_{\mathrm{obs}}$, $V_{\mathrm{DM}}$, $V_{\mathrm{gas}}$, $V_*$ versus the radius $r$ and draw smooth curves through the points (mark your graph as “D2.2”).

Order the contribution of the different components to the observed velocity in descending order. [16]

• Plot uses more than 50% of graph paper: 1.0
• Both axes labels ($V$ and $r$) and their units presented: 2.0
• Ticks and values on axes (or scale written explicitly): 1.0
• All plots labels ($V_{\mathrm{obs}}$, $V_{\mathrm{DM}}$, $V_{\mathrm{gas}}$, $V_*$) presented: 2.0, each label gets 0.5 mark
• Points correctly plotted: 4 (0.1 for each point) Values of $V_{\mathrm{obs}}$, $V_{\mathrm{DM}}$ wrong due to wrong values of $z$ gets a maximum of 4.0 marks
• Smooth curve through points: 4.0 (1.0 per curve)
• Order of contribution: 2.0, correct order only

(D2.3) Take a data points at small $r$ and large $r$ to estimate $\rho_0$ and $r_C$. Note that for large values of $x$, $\arctan(x) \approx \pi/2$ and at small $x$, $\arctan(x) \approx x - x^3/3$. [7]

Solution

Evaluating for $\rho_0$:

$M_{DM}(r) = 4\pi\rho_0 r_C^3[x - \arctan(x)]$, where $x = r/r_C$ [1.0]

$M_{DM}(r) \approx 4\pi\rho_0 r_C^3\left[x - \left(x - \frac{x^3}{3}\right)\right]$, for small $x$ [1.0]

$\rho_0 \approx \frac{2.02\times10^8 M_{\odot}\times3}{4\pi(0.7kpc)^3} = 1.42\times10^8 M_{\odot}/kpc^3$ [1.5]

Selected correct data to put into the formula gets 0.5, correct answer gets 0.5 and correct unit gets 0.5.

Evaluating for $r_C$ (Method 1):

$M_{DM}(r) \approx 4\pi\rho_0 r_C^2\left[r - r_C\frac{\pi}{2}\right]$, [1.0]

Take the last two data points at large $r$, then we get (for $r \gg r_C$

$\Delta M_{DM}(r) \approx 4\pi\rho_0 r_C^2[\Delta r]$ [1.5]

$r_C = 1.01\ kpc$ [1.0]

Evaluating for $r_C$ (Method 2):

$M_{DM}(r) \approx 4\pi\rho_0 r_C^2\left[r - r_C\frac{\pi}{2}\right]$, [1.0]

Take the last data point, we get a cubic equation which students can solve to give

$r_C = -0.855, 0.964, 7.56$ kpc [1.0]

Select the correct $r_C =$ 0.964 for the final answer because for the last data point $r \gg r_C$ [1.5]

Less accurate $r_C$:

Omit the last term,

$M_{DM}(r) \approx 4\pi\rho_0 r_C^2 r$ [1.0]

$r_C \approx \sqrt{\dfrac{1.74\times10^{10}M_{\odot}}{4\pi\times1.40\times\frac{10^8M_{\odot}}{kpc^3}\times12.05kpc}} = 0.901\ kpc$ [1.0]

Part 2 Dark matter distribution

(D2.4) By comparing Equation (4) to a linear function, the central density $\rho_0$ could also be found by a linear fit. Plot an appropriate graph so that a linear fit can be used to find another value of $\rho_0$. Evaluate $\rho_0$ in units of $\frac{M_{\odot}}{kpc^3}$ (mark your graph as “D2.4”). If you cannot find the value of $r_C$ from the previous part, use $r_{\mathrm{C}} =$ 3.2 kpc as an estimate for this part. [19]

Solution 1:

$x = \left[r - r_c \arctan\left(\frac{r}{r_C}\right)\right]$

$y = M_{DM}(r)$

• Choose correct axes: 1.0, each correct axis get 0.5 mark
• Values correctly computed: 8.0 (0.4 for each value, 20 data points)
• Plot uses more than 50% of graph paper: 1.0
• Both axes labels presented: 1.0
• Ticks and values on axes (or scale written explicitly): 1.0
• Points correctly plotted: 4.0 (0.2 for each point, 20 data points) Values of x axis wrong due to previous wrong values get a maximum of 2.0 marks
• Credit for good visual linear fit: 1.0
• Correct value of slope: 1.0
• Correct value of $\rho_0$: 1.0

Solution 2:

$x = \frac{r_C}{r}\arctan\left(\frac{r}{r_C}\right)$

$y = M_{DM}(r)$

• Choose correct axes: 1.0, each correct axis get 0.5 mark
• Values correctly computed: 8.0 (0.4 for each value, 20 data points)
• Plot uses more than 50% of graph paper: 1.0
• Both axes labels presented: 1.0
• Ticks and values on axes (or scale written explicitly): 1.0
• Points correctly plotted: 4.0 (0.2 for each point, 20 data points) Values of x axis wrong due to previous wrong values gets a maximum of 2.0 marks
• Credit for good visual linear fit: 1.0
• Correct value of slope: 1.0
• Correct value of $\rho_0$: 1.0

(D2.5) Compute logarithmic values of the dark matter density, ln [$\rho_{\mathrm{DM}}$(r)], and plot the distribution of the dark matter in the galaxy as a function of radius $r$ on graph paper (mark your graph as “D2.5”). [12]

Solution 1: $r_c = 0.906\ kpc$ from D2.3

• Values correctly computed: 4.0 (0.4 for each values, 10 data points) Values of ln [$\rho_{\mathrm{DM}}$(r)] wrong due to the wrong value of $\rho_0$ gets a maximum of 2.0 marks
• Plot uses more than 50% of graph paper: 1.0
• Both axes labels presented: 1.0
• Ticks and values on axes (or scale written explicitly): 1.0
• Points correctly plotted: 4.0 (0.4 for each point, 10 data points) Values of ln [$\rho_{\mathrm{DM}}$(r)] wrong due to previous wrong values gets a maximum of 2.0 marks
• Smooth curve through points: 1.0

Solution 2: $r_c = 3.2\ kpc$

• Values correctly computed: 4.0 (0.4 for each values, 10 data points) Values of ln [$\rho_{\mathrm{DM}}$(r)] wrong due to the wrong value of $\rho_0$ gets a maximum of 2.0 marks
• Plot uses more than 50% of graph paper: 1.0
• Both axes labels presented: 1.0
• Ticks and values on axes (or scale written explicitly): 1.0
• Points correctly plotted: 4.0 (0.4 for each point, 10 data points) Values of ln [$\rho_{\mathrm{DM}}$(r)] wrong due to previous wrong values gets a maximum of 2.0 marks
• Smooth curve through points: 1.0