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GeCAA 2020, Анализ на данни — Задача 3. Hypervelocity stars

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GeCAA-2020 — Data Analysis · 55 т.

Условие

In recent years, a new field of research has emerged, that of Hypervelocity Stars (HVS for short). These are stars in our Galaxy (mostly at its outskirts), which are moving with excessive velocities and may be escaping from the Milky Way.

In this question, you will use spectroscopic and astrometric measurements in order to calculate the velocity of one such star, called “HVS1”, consider its origin and whether it may escape the Galaxy.

Figure 3 shows a spectrum of HVS1 in the blue to UV part of the spectrum:

Spectrum of HVS1: Flux (erg cm⁻² s⁻¹ Å⁻¹) from about 2×10⁻¹⁷ to 9×10⁻¹⁷ against λ (Å) from 3,600 to 4,500; deep absorption lines labelled H 16, Hη, Hζ, Hε, Hδ, Hγ; weaker lines labelled CaII K (two red arrows mark a double line), HeI 4026, SiII 4130, HeI 4144, MgII 4481.

(a) Determine the spectral type of the star using the standard spectra in the Appendix 3 and the absorption lines identified on the spectrum of HVS1. (Note that the spectrum above contains both stellar and interstellar absorption lines.) [7 т.]

(b) Detailed modeling of the spectral lines places the star between luminosity classes V (Main Sequence) and IV (subgiant).

i. The apparent magnitude of the star in the visual band is $m_V = 19.84$ . Find the absolute magnitude $M_V$ of the star using Appendix 1 for the two possible luminosity classes.

You may ignore the uncertainty in $m_V$ since the uncertainty in your calculation will be dominated by the uncertainty in $M_V$ .

ii. For both these possible luminosity classes, calculate the star’s distance from the Sun, ignoring interstellar absorption.

iii. The galactic coordinates of HVS1 are $l = 227.33537267^\circ$ , $b = 31.33199386^\circ$ . Is the assumption of ignoring interstellar absorption justified? (YES/NO)

iv. The Gaia mission of the European Space Agency has been mapping the Milky Way since 2014, measuring the parallax and proper motion of 1.5 billion stars to an accuracy between 0.04 and 0.1 milliarcseconds (mas). Could Gaia have measured the parallax of HVS1, (A) if it is a MS star? (YES/NO)
(B) if it is a subgiant? (YES/NO)

For the rest of this question, adopt the larger of the distances you have calculated above.

v. Assume that the distance of the Sun from the GC is 8.0 kpc. Make a rough sketch of the relative positions of HVS1, the Sun and the Galactic center (GC). Use it to calculate the distance ( $r$ ) of HVS1 from the GC. [18 т.]

(c) Here, you will calculate the velocity of HVS1.

i. The spectrum in the figure 3 shows two absorption lines due to Ca II. One is caused by the atmosphere of the star and the other is due to the interstellar medium. The shift of this line is due to the motion of the star with respect to the interstellar medium. Measure this Doppler shift and calculate the radial velocity of HVS1 with respect to the Sun.

ii. We are interested in the velocity with respect to the Galactic center. For this, we first need to take into account the velocity of the Sun due to the rotation of the Galaxy. The following equation transforms the velocity of a star of heliocentric radial velocity $v_{r\mathrm{HC}}$ to the one in the galactic rest frame (rf), $v_{\mathrm{rf}}$ :

v_{\mathrm{rf}} = v_{r\mathrm{HC}} + 11.1\cos l\cos b + 247.24\sin l\cos b + 7.25\sin b,

where the speeds are measured in $\mathrm{km\,s^{-1}}$ . Find $v_{\mathrm{rf}}$ for HVS1.

iii. HVS1’s proper motion (milliarcseconds/year) has been measured as:

(\mu_\alpha, \mu_\delta) = (+0.08\pm 0.26,\ -0.12\pm 0.22)\ \mathrm{mas/yr}.

Calculate the tangential velocity component (in km/s) of HVS1. (You may ignore the correction for declination as the star is near the celestial equator).

iv. Calculate the velocity $v_T$ of the star with respect to the Galactic center magnitude in km/s and angle with respect to the direction of the Galactic center in degrees.

v. Assuming this star was born in the Galactic disc, use your calculation of the velocity to estimate where in the Galactic disc it is more likely to have come from:

near to the Galactic center
further out in the Galactic disc [17 т.]

(d) From the energy considerations,

i. Obtain an expression for the escape speed $v_{\mathrm{esc}}$ as a function of the distance from the Galactic center and the enclosed mass.

ii. Calculate the mass of the galaxy (in solar masses) within the radius of the distance of HVS1

M(r) = 4\pi\rho_0 r_c^2\left[r - r_c\arctan\frac{r}{r_c}\right],

where $r_c \approx 8$ pc is a constant of the equation and $\rho_0 = 1.396\times 10^4\ M_\odot\,\mathrm{pc}^{-3}$ .

iii. Calculate the magnitude of the escape velocity at the distance of HVS1.

iv. Is this a runaway star? (YES/NO) [6 т.]

(e) How long has it taken for HVS1 to reach this position? Write answer in years. [2 т.]

(f) On the basis of the spectral type and the luminosity class of this star, estimate the age of HVS1. Compare this with your result in the previous part. Which one of the following statements about the origin of the star is true:

The star was ejected when or shortly after it was formed.
The star was ejected mid-way through its time on the Main Sequence.
The star was ejected towards the end of its time on the Main Sequence. [3 т.]

(g) Astronomers looking for HVS-s start by finding a sample of stars in the Galactic halo which are of a spectral type similar to that of HVS1. Explain why by choosing which one of the following statements is true:

Stars of this spectral type are young and so belong to the native population of the halo.
Stars of this spectral type are old and so belong to the native population of the halo.
Stars of this spectral type are young and so do not belong to the native population of the halo.
Stars of this spectral type are old and so do not belong to the native population of the halo. [2 т.]

Appendix 1

Classification and absolute magnitude of stars ( $M_\mathrm{v}$ )

Sp	Supergiants Ia	Supergiants Ib	Bright giants II	Giants III	Sub-giants IV	Main sequence dwarfs V	ZAMS $^{(a)}$ V	White dwarfs VII	Population II: Sub-dwarfs VI	Population II: Red branch	Population II: Horiz. branch
O5	−6.4			−5.4		−5.7
B0	−6.7	−6.1	−5.4	−5.0	−4.7	−4.1	−3.3	+10.2
B5	−6.9	−5.7	−4.3	−2.4	−1.8	−1.1	−0.2	+10.7			+2.3
A0	−7.1	−5.3	−3.1	−0.2	+0.1	+0.7	+1.5	+11.3			+0.8
A5	−7.7	−4.9	−2.6	+0.5	+1.4	+2.0	+2.4	+12.2			+0.5
F0	−8.2	−4.7	−2.3	+1.2	+2.0	+2.6	+3.1	+12.9			+0.4
F5	−7.7	−4.7	−2.2	+1.4	+2.3	+3.4	+3.9	+13.6	+4.8	+4.8	+0.4
G0	−7.5	−4.7	−2.1	+1.1	+2.9	+4.4	+4.6	+14.3	+5.7	+4.1	+0.3
G5	−7.5	−4.7	−2.1	+0.7	+3.1	+5.1	+5.2	+14.9	+6.4	+2.0	−0.1
K0	−7.5	−4.6	−2.1	+0.5	+3.2	+5.9	+6.0	+15.3	+7.3	−0.2	−0.6
K5	−7.5	−4.6	−2.2	−0.2		+7.3	+7.3	+15	+8.4	−2.2	−2.2
M0	−7.5	−4.6	−2.3	−0.4		+9.0	+9.0	+15	+10	−3	−3
M2	−7		−2.4	−0.6		+10.0	+10.0		+12
M5				−0.8		+11.8	+11.8		+14
M8						+16			+16

Appendix 2

Species and ionization	Identification	Wavelength/nm	Species and ionization	Identification	Wavelength/nm
Ca II	K	393.4	TiO		545–570
Ca II	H	396.8	Na I	D doublet	589.0,589.6
H I	Balmer Hε	397.0	TiO		590–610
Fe I		404.6	CH $_4$	methane band	619
H I	Balmer Hδ	410.1	TiO		620–640
Ca I	g	422.7	H I	Balmer Hα	656.3
CH	G	430–432.5	TiO		665–700
H I	Balmer Hγ	434.0	O $_2$	B (telluric band)	686.7
He II		454.2	H $_2$ O	a (telluric band)	716.0
TiO		475–490	CH $_4$	methane band	725
H I	Balmer Hβ	486.1	H $_2$ O	(telluric band)	732–738
TiO		495–510	O $_2$	A (telluric band)	759.4
[O III]	oxygen	495.9	H $_2$ O	Z (telluric band)	790–840
[O III]	oxygen	500.7	CH $_4$	methane band	798
TiO		515–540	CH $_4$	methane band	887–911

Spectral lines useful in the spectral characterization of low resolution stellar spectra of different stars

Appendix 3

Spectra of main sequence stars of different spectral classes, at the blue end of the spectrum (where there are many absorption lines due to atomic transitions). The spectra have been divided by the respective continuum so that the lines are more easily discernible. The first five panels correspond to MS stars. The sixth panel shows how the lines change for different luminosity classes for the same spectral type.

Panel titled Main Sequence O9 – B5: normalised spectra from 3800 to 4600 (Wavelength) of HD 46202 (O9 V), ω¹ Sco (B1 V), 22 Sco (B2 V), 29 Per (B3 V) and HD 36936 (B5 V); lines labelled H9, H8, Hε, He I, C III, O II, Hδ, He II 4200, Hγ, He II 4542 and the wavelengths 4009, 4026, 4121, 4144, 4387, 4471.

Panel titled Main Sequence B5 – A5: normalised spectra from 3800 to 4600 (Wavelength) of HD 36936 (B5 V), β Per (B8 V), α Lyr (A0 V), β Leo (A3 V) and HD 23194 (A5 V); lines labelled H9, H8, Ca II K, Hε, He I, Hδ, C II, Hγ, Mn I, Ca I, Fe I, Mg II.

Panel titled Main Sequence A5 – G0: normalised spectra from 3800 to 4600 (Wavelength) of HD 23194 (A5 V), HD 23585 (F0 V), HD 27524 (F5 V) and β Com (G0 V); lines labelled H9, H8, Ca II K, Hε/H, Mn I, Hδ, Ca I, Fe I, Hγ, Mg II, and markers Fe I, G-band, Fe I under the G0 V spectrum.

Panel titled Main Sequence G0 – K5: normalised spectra from 3800 to 4600 (Wavelength) of β Com (G0 V), κ¹ Cet (G5 V), σ Dra (K0 V) and 61 Cyg A (K5 V); lines labelled Ca II K, Ca II H, Mn I, Fe I, Hδ, Ca I, Fe I, G-band, Fe I, Hγ, Fe I, Mg II.

Panel titled Main Sequence K5 – M4.5, Normalized Flux: spectra from 3800 to 5600 (Wavelength) of 61 Cyg A (K5 V), 61 Cyg B (K7 V), BD +43 44A (M1.5 V), BD +43 44B (M3.5 V) and o² Eri C (M4.5 Ve); features labelled Ca I, Fe I, MgH, MgH, TiO, Hβ, TiO, Mg b, TiO, CaOH.

Panel titled Luminosity Effects at A0: spectra from 3800 to 4600 (Wavelength) of HR 1040 (A0 Ia), η Leo (A0 Ib), α Dra (A0 III) and α Lyr (A0 Va); lines marked H9, H8, Ca II K, Hε, Hδ, Si II 4128–30, 4172–8, Fe II 4233, Hγ, I.S., Mg II 4481; the Balmer lines are much broader for the lower luminosity classes.

Отговори

Решение

Figure 1 shows a spectrum of HVS1 in the blue to UV part of the spectrum:

Figure 1: The spectrum of HVS1 shifted to the rest frame of the star (i.e., there is no Doppler shift due to the motion of the star along the line of sight).

(a) (7 points) Determine the spectral type of the star using the standard spectra in Appendix 3 and the absorption lines identified on the spectrum of HVS1. (Note that the spectrum above contains both stellar and interstellar absorption lines.)

Solution
By comparing with spectral charts in appendix 3, we notice that

H-lines are the most prominent lines,
He I feature at $4144\ \text{Å}$ is barely visible and
Ca I feature around $4230\ \text{Å}$ is not seen.

Thus, it may be reasonable to restrict the range of spectral types to B5-A5. 2.0
We find that the ratio of the intensity of a pair of lines and compare it with the standard spectra. We see, Ca II/H $\epsilon \sim \frac{1}{4}$ 2.0
For B8 standard spectra it is $\sim\frac{1}{5}$ , for A0 standard spectra it is $\sim\frac{1}{3}$ . 2.0

Thus, we deduce that the spectral type lies somewhere between B8 and A0. One may write any spectral type in the range, say B9. 1.0

(b) (18 points) Detailed modeling of the spectral lines places the star between luminosity classes V (Main Sequence) and IV (subgiant).

i. The apparent magnitude of the star in the visual band is $m_V = 19.84$ . Find the absolute magnitude $M_V$ of the star using Appendix 1 for the two possible luminosity classes.
You may ignore the uncertainty in $m_V$ since the uncertainty in your calculation will be dominated by the uncertainty in $M_V$ .

Solution
Appendix 1 gives absolute magnitude for stars of different spectral and luminosity classes. The relevant rows are A0 and B5. 1.0
In order to estimate the absolute magnitude of a B9 star, interpolate between B5 and A0.

For MS:

\begin{aligned} M(\mathrm{B9}) &= M(\mathrm{A0}) - \frac{(M(\mathrm{A0}) - M(\mathrm{B5}))}{5}\times(10-9)\\ &= 0.7 - \frac{(0.7-(-1.1))}{5} = 0.7 - 0.36\\ &= 0.34 \end{aligned}

2.0

For subgiants:

\begin{aligned} M(\mathrm{B9}) &= M(\mathrm{A0}) - \frac{(M(\mathrm{A0}) - M(\mathrm{B5}))}{5}\times(10-9)\\ &= 0.1 - \frac{(0.1-(-1.8))}{5} = 0.1 - 0.38\\ &= -0.28 \end{aligned}

2.0

ii. For both these possible luminosity classes, calculate the star’s distance from the Sun, ignoring interstellar absorption.

Solution

\begin{aligned} m_v &= M_v + 5\log(d) - 5\\ \therefore d &= 10^{\frac{m_v - M_v + 5}{5}} \end{aligned}

1.0

For MS:

d \approx 79.4\ \mathrm{kpc}

1.0

For subgiants:

d \approx 105.7\ \mathrm{kpc}

1.0

iii. The galactic coordinates of HVS1 are $l = 227.335\,372\,67^\circ$ , $b = 31.331\,993\,86^\circ$ . Is the assumption of ignoring the interstellar absorption justified? Write “YES” or “NO”.

Solution
Yes. HSV1 is off the galactic plane, well into the halo. interstellar absorption should not be very high. Thus, ignoring it will not affect the distance very much. 2.0

(A) if it is a MS star? Write “YES” or “NO”.

(B) if it is a subgiant? Write “YES” or “NO”.

Solution

(A) if it is a MS star,

\pi = 1/d \sim 0.01\mathrm{mas} < \pi_{\mathrm{GAIA}}

Hence, the answer is “No”. 1.0

(B) If it is a subgiant, then the distance is even larger. Hence, the answer is again “No”. 1.0

For the rest of this question, adopt the larger of the two distances you have calculated above.

v. Assume that the distance of the Sun from the Galactic center is $8.0\ \mathrm{kpc}$ .
Make a rough sketch of the relative positions of HVS1, the Sun and the Galactic center. Use it to calculate the distance ( $r$ ) of HVS1 from the Galactic center.

Solution

Two sketches. Left, to scale: a long line from HVS1 to the nearby points Sun and GC. Right, details: points H (HVS1) above P (its projection on the Galactic plane), S (Sun) and C (Galactic center); the vertical HP, lines HS, HC, PS and a dashed PC; the latitude b and longitude l marked by arcs at S, with red axis arrows at S.

The left panel is to the scale. The right panel shows details of the drawing. 2.0

\begin{aligned} d(HC)^2 &= d(HP)^2 + d(PC)^2\\ &= d(HP)^2 + \left[d(SC)^2 + d(PS)^2 - 2\cdot d(SC)\cdot d(PS)\cos(\measuredangle PSC)\right]\\ r^2 &= (d\sin b)^2 + r_\odot^2 + (d\cos b)^2 - 2r_\odot d\cos b\cos(360^\circ - l)\\ \therefore r &= \sqrt{d^2 + r_\odot^2 - 2r_\odot d\cos b\cos l}\\ &= \sqrt{105.7^2 + 8^2 - 2\times 8\times 105.7\cos(31.332^\circ)\cos(227.335^\circ)}\\ \therefore r &\approx 110\ \mathrm{kpc} \end{aligned}

4.0

i. The spectrum in Fig1 shows two absorption lines due to Ca II. One is caused by the atmosphere of the star and the other is due to the interstellar medium. The shift of this line is due to the motion of the star with respect to the interstellar medium. Measure this Doppler shift and calculate the radial velocity of HVS1 with respect to the Sun.

Solution From the spectrum, we can estimate the line shift as

\Delta\lambda = (11\pm 2)\ \text{Å},\ \ \lambda_0 = 3934\ \text{Å}

3.0

Doppler shift gives the heliocentric velocity

v_{\mathrm{rHC}} = \frac{c\Delta\lambda}{\lambda_0} = (840\pm 150)\ \mathrm{km\,s^{-1}}

2.0

ii. We are interested in the velocity with respect to the Galactic center. For this we first need to take into account the velocity of the Sun due to the rotation of the Galaxy. The following equation transforms the velocity of a star of heliocentric radial velocity $v_{\mathrm{rHC}}$ to one in the Galactic rest frame (rf), $v_{\mathrm{rf}}$ :

v_{\mathrm{rf}} = v_{\mathrm{rHC}} + 11.1\cos l\cos b + 247.24\sin l\ cosb + 7.25\sin b,

where the speeds are measured in $\mathrm{km\,s^{-1}}$ .
Find $v_{\mathrm{rf}}$ for HVS1.

Solution

\begin{aligned} v_{\mathrm{rf}} &= v_{\mathrm{rHC}} + 11.1\cos l\cos b + 247.24\sin l\ cosb + 7.25\sin b\\ &= 840 + 11.1\cos(227.335^\circ)\cos(31.332^\circ)\\ &\quad + 247.24\sin(227.335^\circ)\cos(31.332^\circ) + 7.25\sin(31.332^\circ)\\ v_{\mathrm{rf}} &= (680\pm 130)\ \mathrm{km\,s^{-1}} \end{aligned}

2.0

iii. HVS1’s proper motion has been measured as:

(\mu_\alpha, \mu_\delta) = (+0.08\pm 0.26, -0.12\pm 0.22)\ \mathrm{mas/yr}.

Calculate the tangential velocity component (in $\mathrm{km\,s^{-1}}$ ) of HVS1. (You may ignore the correction for declination as the star is near the celestial equator).

Solution
The length of an arc does not depend on the coordinate system. Thus,

\begin{aligned} \Delta\mu &= \sqrt{\mu_\alpha^2 + \mu_\delta^2}\\ &= \sqrt{0.08^2 + 0.12^2}\\ \Delta\mu &= (0.144\pm 0.340)\ \mathrm{mas/yr} \end{aligned}

2.0

\begin{aligned} v_\theta = d\Delta\mu &= \frac{105.7\times 10^3\times 3.086\times 10^{13}\times 0.144\times 10^{-3}}{206\,265\times 3.16\times 10^7}\\ &= (72\pm 170)\ \mathrm{km\,s^{-1}} \end{aligned}

2.0

iv. Calculate the velocity $v_{\mathrm{T}}$ of the star with respect to the Galactic center (magnitude in $\mathrm{km\,s^{-1}}$ and angle with respect to direction of the Galactic center).

Solution
The direction to the Galactic center is also perpendicular to the plane of the sky. Thus,

\begin{aligned} v_{\mathrm{T}} &= \sqrt{v_{\mathrm{rf}}^2 + v_\theta^2} = \sqrt{680^2 + 72^2}\\ v_{\mathrm{T}} &= (684\pm 210)\ \mathrm{km\,s^{-1}} \end{aligned}

2.0

i.e. it is almost radial.

\theta = \tan^{-1}\left(\frac{v_\theta}{v_{\mathrm{rf}}}\right) \approx \tan^{-1}(0.105) \approx 6^\circ

2.0

v. Assuming this star was born within the Galactic disc, use your calculation of the velocity to estimate where in the Galactic disc it is more likely to have come from:

(A) near to the Galactic center

(B) further out in the Galactic disc

Solution As the velocity is almost radial, extrapolating backwards, the star would be in galactic disk somewhere close to the galactic centre. Thus, the answer is A. 2.0

(d) (6 points) From the energy considerations,

i. Write expression for the escape speed $v_{\mathrm{esc}}$ as a function of the distance from the Galactic center and the enclosed mass.

Solution

v_{esc} = \sqrt{\frac{2GM}{r}}

1.0

ii. Calculate the mass of the galaxy (in solar masses) within the radius of the distance of HVS1.

M_r = 4\pi\rho_0 r_c^2\left[r - r_c\tan^{-1}\left(\frac{r}{r_c}\right)\right]

where $r_c \approx 8\ \mathrm{pc}$ is an constant of the equation and $\rho_0 = 1.396\times 10^4 M_\odot\mathrm{pc}^{-3}$ .

Solution

\begin{aligned} M_r &= 4\pi\rho_0 r_c^2\left[r - r_c\tan^{-1}\left(\frac{r}{r_c}\right)\right]\\ &= 4\pi\times 1.396\times 10^4\times 8^2\left[110 - 8\times\tan^{-1}\left(\frac{110}{8}\right)\right]\\ M_r(110\ \mathrm{kpc}) &\approx 1.2\times 10^{12} M_\odot \end{aligned}

2.0

iii. Calculate the magnitude of the escape velocity at the distance of HVS1.

Solution

\begin{aligned} v_{esc} &= \sqrt{\frac{2GM}{r}} = \sqrt{\frac{2\times 6.674\times 10^{-11}\times 1.2\times 10^{12}\times 1.998\times 10^{30}}{110\times 10^3\times 3.086\times 10^{16}}}\\ &= 311\ \mathrm{km\,s^{-1}} \end{aligned}

2.0

iv. Is this a runaway star? Write “YES” or “NO”.

Solution
$v_{\mathrm{rf}} > v_{esc} \implies$ Yes. 1.0

(e) (2 points) How long has it taken for HVS1 to reach this position?

Solution If HSV1 originated close to the galactic center and assuming is was ejected at the measured speed, it has been traveling for

t_{\mathrm{trav}} = \frac{r}{v_{\mathrm{rf}}} = \frac{110\times 10^3\times 3.086\times 10^{16}}{680\times 10^3} = 1.6\times 10^8\ \mathrm{yr}

2.0

(f) (3 points) On the basis of the spectral type and the luminosity class of this star, estimate the age of HVS1 and compare this with your result in the previous part. Which one of the following statements about the origin of the star is true:

(A) The star was ejected when or shortly after it was formed.

(B) The star was ejected mid-way through its time on the Main Sequence.

Solution
Main Sequence life time for the star will approximately be

t \sim \frac{M}{L} \sim L^{-5/7}

Comparing with absolute magnitude of the Sun,

t \sim 10^8\ \mathrm{yr}

This is very similar to the time taken by the star to travel from the disk to present position. Thus, the answer is A. 3.0

(g) (2 points) Astronomers looking for HVS-s start by finding a sample of stars in the Galactic halo which are of a spectral type similar to that of HVS1. Explain why by choosing which one of the following statements is true:

(A) Stars of this spectral type are young and so belong to the native population of the halo.

(B) Stars of this spectral type are old and so belong to the native population of the halo.

(D) Stars of this spectral type are old and so do not belong to the native population of the halo.

Solution Stars of early type are very young, so they cannot belong to the native population of the halo. They must be coming from elsewhere. As a result, they may be moving very fast and some of them may be on their way to abandoning the Milky Way.

$\implies$ C. 2.0

Оригинал в Архива: GeCAA-2020_Data_Analysis.pdf · официални решения: Data Analysis Solutions.pdf