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The reason for this research is to unambiguously fit the spectral line of neutral-hydrogen in a spiral galaxy. Much research has been done relating to fitting the spin-flip of neutral-hydrogen in spiral galaxies, but they do not actually show the fit of the spectral line in a significant number of channels. Markov Chain Monte Carlo modelling will be used to find error in the 3 parameters of a Gaussian fit (mean, peak amplitude and the variance of each channel). No polynomial fit of Gauss-Hermite line profiles of the channels have been done for a significant number of channels lately in the previous researches but in this research, we will be using a large number of channels to find a high quality fit of a spectral line with together with Gauss-Hermite profiles.


Radio astronomy is an important technique used by astronomers to study the universe. Measurements of radio spectral line emission have identified and characterised the birth of stars and galaxies (Miller, 1998). For us to be able to study the universe we need telescopes. A radio telescope is a telescope that is designed to receive radio waves from space. How does a telescope work? One or more antennas are needed to collect incoming radio waves; the receiver or amplifier boosts the weak radio signal to a measurable level. A recorder will then record the signal and store to computer memory where it will be analysed. One of the first early telescopes was the Dover Heights in Sydney Australia and from then more telescopes were built, Parkes radio telescope is also one of the early telescopes. Interferometers, using multiple connected dishes, allow astronomers to study objects in greater detail than in using a simple dish. More, larger and bigger telescopes were built over the years. Which brings us to the fact that the larger the telescope the larger the total collecting area. This gave birth to the idea of the Square Kilometre Array (SKA) and it is the next generation radio interferometer telescope to be built jointly in Southern Africa and Australia. By SKA we mean that there will be 1 square kilometre of collecting area, which requires about 5000 dishes with 16m diameter. The collecting area of the dish can be found using the formula;
Where N is the total number of dishes and r is the radius of the dish.
It began in 2013 with seven dishes at the central core in the Karoo desert of South Africa, where it is known as the KAT7.
The pathfinder for the SKA in South Africa is called MEERKAT which had only 16 dishes in 2016 and was formally unveiled and started taking data using its full 64 dishes in July 2018. MeerKAT has a maximum baseline of 8km.

There are three types of radio emissions being;
Continuum sources(non- thermal)
Spectral line sources
The blackbody sources are the cosmic microwave background requires low temperatures of;
?T=0.2898cmK. (1)
Continuum emissions are non- thermal and can be defined as emissions at all radio wavelengths where by a charged particle moving in an electromagnetic field experiences acceleration and emits a photon. Continuum emissions are from the following sources; radio emission lines, the neutral hydrogen line (HI line), refers to the electromagnetic radiation spectral line that are produced by change in energy states of neutral hydrogen states. Spin flip transitions occur, which is the frequency of the quanta that are emitted by transition between two energy levels giving us the following wavelength:
?=1/v×c=21.106cm (2)
Where v is the frequency and c is the speed of light,
Hence the name 21 cm neutral hydrogen line. With this information we can now talk about our galaxy what kind of Galaxy it is and what kind of emission lines it emits.

NGC 2403 is an intermediate low density optical spiral galaxy with ongoing star formation. Its distance has been measured from analysis of Cepheid variables to be 3.2 Mpc that is approximately 8 million light years (W.J.G. De Blok, 2008)Since it has on going star formation it has HII (ionised hydrogen) regions. It has a right ascension of about 07 36 51.1 hours. The bulge of this galaxy is slightly elongated therefore it is called a barred galaxy. Star formation is active in the spiral disk, its spiral arms have dark bands formed by interstellar dust therefore there is a large amount of neutral hydrogen gas responsible for the formation of stars. It is a late-type Sc spiral galaxy and a member of the M81 group. The spiral structure and the stellar content closely resemble those present in M33 according to (Filippo Fraternali, 2002). NGC 2403 clusters are too far to allow the observation of a clear resolution, the number of clusters is too small. The reddening of NGC 2403 can be taken to be (B – V) = 0.037 magnitude.

Figure 1: Shows the major axis luminosity density (surface brightness) for NGC 2403. The profile is decomposed into bulge (short and long dashed) and disk (short dashed) components. (Kent, 1986) Lower panels: Observed rotation curves (pulses) and best fitting full solution (solid line). The separate contributions of the disk and dark matter halo are drawn.

Figure 2: NGC 2403 image taken from The Subaru Galaxy B band (0.45?m: blue), R (0.65?m: green), IA651 (0.651?m: red) false color image.

The reason why we want to fit the spectral lines is that galaxies rotate. This means that there is a different Doppler shift across the galaxy depending on the inclination, position angle, radius from the centre, etc. If we can measure that rotation speed then we can use Newton’s laws to determine the mass of the galaxy and hence how much Dark Matter there is.
What we see in a spectrum (lines, line shapes and ionisation states) for gas in a galaxy is the shape of the line profile which is determined by random motions which may not be isotropic, gas density and the circular velocity at the radius of the gas cloud. The fit spectral line gives us the rotation speed of the gas in the galaxy which gives the total mass of the galaxy. The equation below gives a rough connection between the circular velocity and gravity of the galaxy.
V_c= ?((GM(r))/r) (3)
Different models can be used to fit the spectra line emission of neutral hydrogen in the galaxy. The commonly used method to study gaseous distribution and kinematics on the fitting of emission-line profiles is by Gaussian curve or the Gaussian model. We also make use of Markov Chain Monte Carlo (MCMC) methods, Monte Carlo methods or simulations that simulate a random physical process in order to estimate something about the outcome of that process and they use computational algorithms that rely on random sampling (repeated) to obtain numerical results. The importance of the MCMC is to calculate sums over states which are not easily computed. In simple terms if we want to choose a certain set of states for the sum, we will need to generate a string of states of one another thus employing the Markov Chain (Newman, 2013).

The general form for a Gaussian is
G= 1/?(2??^2 ) e^((-?(x- ?)?^2)/(2?^2 )) (3) , where
We want to investigate whether the pure Gaussian model is insufficient to fit the spectral lines because they are typically skewed and lopsided. Therefore, we use Gauss- Hermit polynomials as additional models for comparison.

The general form of Gaussian-Hermite polynomial is given by:
GH= (A?(?))/?×?_(j=0)^n??h_j H_j (?) ? (4)
Where A is the amplitude,
?= (?- ?_c)/? (5)
?(?)= 1/?2? e^((-?^2)?2) (6)
For n = 4
GH=A?(?)/? 1+h_3 H_3 (?)+ h_4 H_4 (?) (7)
H_3= 1/?6 (2?(2?^3 )-3?2? ) (8)
H_4= 1/?24( 4?^4-12?^2+3) (9)

When h_3= h_4=0 a plain Gaussian distribution formula is what we get. ?_c,?,h_3, h_4 are just fitting parameters where ?_c is the central wavelength and ? is the velocity dispersion. The h_3 Gauss-Hermite moment measures asymmetric deviations from a Gaussian model profile while the h_4 moment quantifies the peak of the profile with h_4; 0 for a more peaked and h_4 ; 0 for a broader peaked profile. We naturally expect a Gaussian-Hermite model gives a better fit to the profile since it has more parameters and more flexibility than the Gaussian model.
We make a comparison between the GHP model and the spectrum in each pixel for the galaxy by using Markov Chain Monte Carlo (MCMC) to jump through the multidimensional parameter space and use chi square to decide on the goodness of fit.
?^2=?_(j=1)^n??data(k,i,j)-Ae^((((-(k- ?)^2)??^2 ))??noise?^2 ) ? (10)
The chi square should be reduced to approximately 1 (that is?^2/(number of points)) for a good fit.
The difference between the Gauss-Hermite polynomials H_3 and H_4 is that the H_4 has more parameters and therefore has more flexibility than H_3 polynomial fit. A good of the spectral line fit gives a better estimate of the rotation speed and thus the mass of the galaxy.

Finding the right spectral line fit model
The data used was from the Very Large Array (VLA) radio inferometer telescope in central New Mexico America. It is a cm- wavelength radio astronomy observatory, it comprises of 27 25-meter radio dishes made in a Y-shape array and all its equipment and computing power function as the inferometer (Catherin, 2017).

Figure 3: Image for the VLA telescopes adapted from the savy magazine
It has a collecting area of 13 250 square meter (far less than the 1 000 000 m^2 of the SKA) with an angular resolution of about (0.2, 0.04) arc seconds. To analyse the data we wrote from scratch a python program. The data was in the standard format flexible image transport system or FITS. Data was extracted from the file ngc2403.fits, ngc2403 being the name of the galaxy that was analysed and fits standing for the file type. The data was in 3-d (3-dimensional that is in x, y, z components where z represents thickness of the galaxy, and x, y are the image we see). Our objective was to fit different GHP models mentioned to the spectrum in each of the pixels to find the best fit spectral line model and its uncertainty.

Figure 4: The 3-d image of the data

Before fitting the GHPs, we simply read in the above image to see what our galaxy looks like. We found the maximum intensity of emission in each pixel and plotted it (see above figure) giving us a two-dimensional picture of the ngc2403 galaxy. One can clearly see that there is a spiral structure associated with the neutral hydrogen in the galaxy and that there is very little neutral hydrogen at the centre of the galaxy. The reason for this could be there could be feedback (UV emission or supernovae) from stars in the centre of the galaxy.

Next we made a plot that gave us a figure showing the red and blue shifts of our galaxy, where we used the spectral channel with maximum emission to encode the magnitude of the Doppler effect. A channel is simply the frequency of the galaxy. Clearly you can see that to the top left of the galaxy there is a blue shift and the bottom right is there is a red shift. The unclear colours that surround black line on the figure below we consider to be noise signal.

Figure 5: NGC2403 galaxy’s red shift and blue shift image.
The possible amplitude was found using the formula;
A=max?(data(i,j,k))×?(2??^2 ) (13)
Then determined the possible best values for chi using the formula:
X^2=???data(i,j,k)-A? e^(-((?k-?)?^2)?(2?^2 )) (14)
Where ? and ? were assigned any values. Their real values were found by using the likelihood theory. The likelihood theory was also used to find the best values of the amplitude.
The Gaussian distribution of the data was found from the formula below;
G=A/?(2??^2 ) e^(-?(k- ?)?^2?(2?^2 )) (15)
Which is just the re-make of equation (3)

Plots for the spectral fit of the Gaussian was made using the best values of the amplitude, ? and ? and chi against the channels but the spectral fit appeared skewed in simple terms it was not a perfect fit, so we were to use a different model. Therefore, we employed the Gaussian-Hermite models to find perfect spectral fit the Gaussian-Hermite with the polynomials against the channels using equations (3) to (9).

Tilted rings Model
From the above image were able to estimate the centre of the galaxy, inclination, inclination angle and the systemic velocity of the galaxy. With the above were able to find the Cartesian coordinates x and y position of the galaxy using the formula
x=x_0-radius×(cos?(?)×cos?(i)×sin?(?)+cos(?)×sin?(?)) (11)
y=y_0+radius×(cos?(?)×cos?(?)-cos?(i)×sin?(?)×sin?(?)) (12)
? was taken to be from 0 to 2?.
? is the inclination of the galaxy which is just the rotational angle of the rotational velocity and,
? is the position angle of the galaxy and they are values we approximated also.

The velocity model was used to calculate the velocity of the galaxy using the formula;
Vm=Vsys+V(test)×(sin?(i)×cos?(?) ) (16)
V_sys Is the systemic velocity.
V(test) is the model values of the circular velocity.
A chi square fitting for the model was done using equation (10) as:
?^(2 )=?_(i=1)^(N=62)?(?V(??_i)- Vm(Vc,?,i,?_j))/???(??_j)?^2 (17)
Because we want to find the best fit the tilted model ring of HI (neutral hydrogen) that is find the maximum of the local part of the parameter space. So we found the likelihood of chi square using equation (10) using the theory that if approximately equal to 1 it is a good fit, thus getting better values of the chis, test velocities and of the inclination angle.
We made ring plots of chis and the velocity model with respect to the centre positions just to get the idea on what kind of rings we should have and if we have the right approximated rotational velocity of the galaxy.

Figure 6: Gives an idea on how the tilted for the galaxy should be like

Pure Gaussian fit

Figure 6: Shows the pure Gaussian fit of the parameters ,

Gaussian-Hermite fit with H3 polynomial

Figure 7: Shows the H_3 Gauss- Hermite polynomial fit

Gaussian-Hermite fit with H_4 polynomial

Figure 8: Shows the H_4 Gauss-Hermite polynomial fit

Gaussian-Hermite fit with both H_3 and H_4 polynomials

Figure 9: Gauss-Hermite polynomial fit for both H_3 and H_4
In the above figures; figure 6, figure 7, figure 8 and figure 9, the first plot shows a plot of the amplitude against the likelihood which we get using the chi square and Monte Carlo Markov Chain Methods. The second plot shows a plot of the mean average of the data (channels) against the likelihood while the third plot shows a plot of the standard deviation of the channels against the likelihood chi. The next plot shows a plot of the emissions or the Gaussian fit of the spectral line of the hydrogen emissions, the small bars are a plot of the noise to the points of the channels against the data and since the noise value is very small the error bars are not clearly seen. The least but not last plot is a plot of the H_3 polynomial against the best chi values and the last plot shows the plot for H_4 polynomial against the best value of the likelihood. This means the a good ?_c which is the central wavelength ranges from 15-35. The uncertainty value is approximately 0.1.
By preforming various plots of the data to find where the spectra line perfectly fits the models we have found that from the thin slice we have done across the data the spectra line fits well between the 70 and 75 points of the data. In the above figure a number of jumps were made to give a clear picture of where the amplitude, mean, standard deviations of the data exist.

Comparisons for the spectral line fit models

Figure 10: Shows the comparisons between the spectral line fit models
In the figure above the first plot is the spectral line fit for the Gaussian-Hermite H_4 polynomial which ranges from.If one takes a closer look at the spectral line fit it appears not to be perfect the same goes for the second graph which is the spectral line fit for the pure Gaussian model. The fourth graph which is for the Gaussian-Hermite H_3 also does not show a perfect spectral Line fit. This is due to the fact that the spectra line is not a symmetric function. The fact is that galaxies are full of dark matter therefore we need the best possible models of the galaxy to measure how fast the galaxy rotates to compare with the created models. The last graph we did not mention from figure 10 is the Gaussian-Hermite model for both H_3 and H_4 polynomials which shows a good spectral line fit. The reason being that the Gaussian-Hermite polynomial with more parameters would give a better fit. The Gaussian-Hermite account for asymmetry of the emissions well giving us a reason to believe that the emission of the neutral hydrogen is just within a certain region and where there is no emission which gives reason to believe that it is dark matter that exist within a galaxy.
We can also compare the mean values of the different Gaussian-Hermite plynomials mean profiles.

Figure 11a: Mean channel of pure Gaussian Figure 11b: Mean channel of Gaussian Hermite H4 polynomial

Figure 11c: Mean channel of Gaussian- Figure 11d: Mean Channel of Gaussian-
Hermite H3 and H4 polynomial Hermite H3 polynomial
The mean channel of the above figure shows the rotational speed of the galaxy. Referring to the above figures the one with a smooth normal curve is the mean channel for the pure Gaussian which means gives better approximations of the rotational velocity of the galaxy.
In figure 6 there is something we did not explain about why the last two graphs in the figure just show straight lines. That is because the model is a pure Gaussian, therefore there are no values for H3 and H4. The same reason applies for the 6th and 5th graphs in figure 7 and 8 respectively. Figure 7 has no values for the h4 polynomial whilst figure 8 has no values for h3 polynomial. From figure 6-8 we account for the noise which are those error bars we see in the fourth graph in the mentioned figures and it appears that the noise value is very small. The reason why we account for the noise is that in all the data we gather there is always Poissonian noise which is about?(?10?^4 ).

Looking at the results we have found we can conclude that the Gaussian-Hermite polynomials are a better model to fit to the spectra line of a neutral-hydrogen galaxy like NGC 2403 with an uncertainty of 0.1. There is always the presence of dark matter which accounts why the Spectra line is not symmetric and in the region there is no ongoing star formation.

I would recommend this project for someone who would like to research more about the NGC2403 galaxy and even go at arm’s length to research more about other galaxies as well. The results for the tilted rings are not shown this is because it is still a work in progress which one could also embark further on to find the rotational curves of the galaxy.

Catherin, A. O., 2017. The Very Large Array Telescope. A world class Observatory, 16 November.
Filippo Fraternali, G. V. M. R. S. T. O., 2002. Deep survey of the spiral galaxy NGC2403. The Astronomical Journal, Issue 123.
Kent, S. M., 1986. Dark Matter in Spiral Galaxies Galaxies with optical rotation Curves. Astronomical Journal, 91(1301-1327).
Miller, D., 1998. Basic of Radio Astronomy for the Goldstone-Applevalley Radio Telescope. California: Jet Propulsion Laboratory, California Institute of Technology.
Newman, M., 2013. Random Processes and Monte Carlo Methods. In: Computational Physics. USA: University of Michigan, pp. 444-479.
W.J.G. De Blok, F. W. E. B. C. T. S.-H. O.-H. R. C. K. J., 2008. High Resolution Rotation Curves and galaxy mass models from things. 2(2100), p. 105.

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