Q1-1

Data Analysis

English (Official)

Data Analysis 1: Scaling Relations (75 points)

Please read the general instructions in the separate envelope before you start this problem.
Spiral galaxies are disk-like rotating structures, whose dynamical state is fairly grasped by the so-called
rotation curves, quantifying the mean rotational velocity of the disk at different distances from the center
(see Figure 1, curve B). An interesting feature is the flat region of the curve, which is attributed to the
presence of dark matter. Without it, rotation velocities would drop steadily at large radii from the center,
as depicted in curve A.

Figure 1: Rotation curves. Circular velocity (Y-axis) vs Radius (X-axis)

In disk galaxies a strong correlation has been observed between the intrinsic luminosity of the whole
galaxy and the asymptotic rotational velocity (as given by the rotation curve for the outer edge of the
galaxy i.e. 𝑅max ), a result that is known as the Tully-Fisher relation. This relation also holds if you use
the luminosity in a specific band. This is shown on Figure 2 for a number of galaxies in a galaxy cluster.
Every dot is a galaxy, and the solid line is the best-fit linear relation between absolute magnitude in 𝐾
band and 𝑙𝑜𝑔10 (𝑉max ) for the whole sample.

Figure 2: Absolute magnitude in 𝐾 band vs log10 (𝑉max [𝑘𝑚𝑠−1 ]). Tully-Fisher relation for several
galaxies. Every dot represents a galaxy. The dark points are five selected galaxies, for which
we will provide some numbers in part 1.2.
 Q1-2
Data Analysis

English (Official)

Figure 3: Gas fraction vs stellar mass.

Another interesting trend is shown in Figure 3: disks with larger stellar masses (𝑀∗ ) tend to have smaller
gas fractions (𝑀𝑔𝑎𝑠 /𝑀∗ ).
In the following questions you will be asked to extract physical information about the galaxies using the
scaling relations just introduced. Consider the following guidelines:
• Assume that 𝑉max was measured at the same radius for all galaxies (𝑅max ), in the flat part of the
rotation curves and well beyond the end of the stellar disk.
• Use 𝑀𝑑𝑚 for the dark matter mass up to 𝑅𝑚𝑎𝑥 and 𝑀𝑡𝑜𝑡 for the sum of all components.(gas, stars
and dark matter)
• Assume that all galaxies have identical stellar populations1 , and assume that the gaseous compo-
nent does not interact with the stellar light. .
• The galaxy cluster is far away. Its distance is much larger than the cluster size.
• In spherically-symmetric mass distributions, to infer the gravitational effect on a particle at distance
𝑟 from the center, it suffices to consider the total mass enclosed up to that radius 𝑀 (≤ 𝑟) as if it
were placed at the very center of the distribution.
 Q1-3
Data Analysis

English (Official)

1
The term stellar population refers to the type of stars that are present in a galaxy, and the relative
amount of each type with respect to the total number of stars.

Part 1 (20 points).

1.1 From an analysis of Figure 3, find the appropriate constants in the following 5.0pt
relation: 𝑀𝑔𝑎𝑠 = 𝑎 × 𝑀∗𝑏

𝑎=?

𝑏=?

1.2 In the plot of the Tully-Fisher relation there are 5 highlighted points. Data for 15.0pt
these 5 galaxies is given in the following table. Use this dataset to find the
appropriate constants for TF relation presented below the table, by means of a
linear fit using the method of least squares.
 
Note: Treat 𝑙𝑜𝑔10 (𝑉max ) as the 𝑥 variable and 𝐾 as the 𝑦 variable in the linear fit.

𝑉max [𝑘𝑚/𝑠] 𝐾[𝑚𝑎𝑔]

79.4 −16.8
100.1 −19.2
158.5 −21.3
251.2 −21.4
316.2 −24.0

𝐾 = 𝑐 × 𝑙𝑜𝑔10 (𝑉𝑚𝑎𝑥 ) + 𝑑

𝑐=?

𝑑=?

Part 2 (16 points).

For two galaxies, G1 and G2, in the cluster, the recorded 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 magnitudes are:

𝑘1 = 19.2 ; 𝑘2 = 25.2

Using this information and the relations calibrated in Part 1 find the correct exponents in the following
equations:

2.1 6.0pt
𝑀∗1
= 10𝑒 ; 𝑒=?
𝑀∗2
 Q1-4
Data Analysis

English (Official)

2.2 4.0pt
𝑀𝑔𝑎𝑠1
= 10𝑓 ; 𝑓 =?
𝑀𝑔𝑎𝑠2

2.3 6.0pt
𝑀𝑡𝑜𝑡1
= 10𝑔 ; 𝑔=?
𝑀𝑡𝑜𝑡2

Part 3 (15 points).

3.1 15.0pt

Apparent
Galaxy 𝑀𝑔𝑎𝑠 [𝑀⊙ ] 𝑀∗ [𝑀⊙ ] 𝑀𝑑𝑚 [𝑀⊙ ] 𝑀𝑡𝑜𝑡 [𝑀⊙ ]
magnitude 𝑘

𝐺1 19.2 4.39×1011

Fill in the missing values in the table using the fact that for galaxy 𝐺1 , the dark-
to-baryonic mass ratio up to 𝑅max is 6.82.

Part 4 (24 points).

4.1 Consider a systematic uncertainty of 𝜎𝑠𝑦𝑠 = ±0.2 in each apparent magnitude 4.0pt
due to CCD calibration errors. Then 𝑘1 must be read as 𝑘1 = 19.2 ± 0.2, i.e., the
only thing we know is that 𝑘1 most likely lies in the interval [19.0, 19.4]. The same
goes for 𝑘2 .
Recalculate the exponent in the scaling relation 𝑀 𝑀∗2 = 10 (found in 2.1), ex-
∗1 𝑒

pressing 𝑒 as an interval estimated by considering the extreme possible varia-

tions in 𝑘1 and 𝑘2 .

𝑒 ∈ [?, ?]
 Q1-5
Data Analysis

English (Official)

4.2 Now we consider that there is always a natural spread of the data around any 10.0pt
relation. For instance, for a given value of the 𝐾 magnitude the TF relation
gives a single value of 𝑙𝑜𝑔10 (𝑉max ), but it would be more realistic to report an
interval of plausible values, derived from the natural spread of the data around
the mean TF relation. We call this the statistical uncertainty, 𝜎𝑠𝑡𝑎𝑡 .
Estimate the statistical uncertainty if 𝑙𝑜𝑔10 (𝑉max ) is inferred from 𝐾 using the
TF relation from question 1.2. For this, consider for each point the difference
between the value of 𝑙𝑜𝑔10 (𝑉max ) estimated from 𝐾 using your linear fit and the
actual measurement of 𝑙𝑜𝑔10 (𝑉max ), and take 𝜎𝑠𝑡𝑎𝑡 as two times the root mean
square (RMS) of these differences† .

𝜎𝑠𝑡𝑎𝑡 = ?
†
The RMS of a set of values is the square root of the arithmetic mean of the
squares of those values.

4.3 Recalculate the exponent in the scaling relation 𝑀𝑀𝑡𝑜𝑡2 = 10 , expressing 𝑔 as an

𝑡𝑜𝑡1 𝑔
10.0pt
interval estimated by considering the extreme possible variations arising from
both the systematic and statistical uncertainties:

𝑔 ∈ [?, ?]