Kinematics of two spiral galaxies
Kinematics of two spiral galaxies Pag. 5
The sky as a laboratory / 2011INTRODUCTION AND PURPOSES OF THE RESEARCH
The motion of a spiral galaxy is made by two contributes: the bulge has the movement of a rigid body and the disc, on the contrary, has a Keplerian movement. The purpose of our research is to represent experimentally the rotational curve of a spiral galaxy that describes how velocity changes in connection with the distance from the galactic centre. Therefore the model that we want to obtain must, near to the centre, describe the circular uniform motion of a rigid body (that respects the law v=ω*r, in which ω is the angular velocity and r the distance from the centre). The graphic must be a straight line. Instead a second part must, on the contrary, describe the Keplerian motion plus the contribute, found experimentally, of the dark matter (that respects the law v2*r=constant and ), in this part the graphic must be similar to the movement of a inverse quadratic proportion (obviously considering that the result, not exactly the attended from an inverse quadratic proportion, is explainable only considering the influence of the dark matter). Our second achievement is to determinate the total mass of the galaxy using the law known as the Theorem of the Viriale (U+2K=0).
TECHNICAL NOTE
In order to determinate the recessional velocities or the rotation velocities of the galaxies at different distances from the centres, we use the Doppler effect, considering that if we have a moving source, the wavelength perceived will be different from the original emitted wave obtained in laboratory.
OBSERVATIVES DATA
Our analysis has been lead starting from the spectra of two galaxies, NGC2336 and NGC2841, acquired with the telescope of Asiago Monte Ekar, with a medium posing time of about 300 seconds; the instruments that we used for the analysis had been this programs: “IRAF”, “Topcat” and “ds9”.
The acquired spectra:
NGC2336
NGC2841
FIRST OPERATIONS
First of all we needed to determinate the positions of the centres of the galaxies, indeed in over there rotational velocity is null because it’s obtained by this formula: v=ω*r and in the centres r=0, so v=0. To be able to determinate the positions (coordinates in pixels) of the centres, we considered the part of the spectra where we see the continuous and using the command implot, we obtained this results:
X / YNGC2336 / 1542 / 243
NGC2841 / 1468 / 241
The values associated to the top of the emission peaks are the values to which we assigned the value r=0, that we would consider to calculate the distances from the centres valued in parsec.
We proceeded “plotting” the part of the spectra where we saw the emission line Hα of hydrogen, we wanted indeed define each wave length perceived while we progressively go away from the galactic centre. We noticed however that Hα was not always “well visible” near to the continuous where it was often absorbed, so for both of the galaxies we decided to use, for our research, the forbidden line [NII] of nitrogen. With the software ds9 we individuated the intervals of wave lengths in which we saw the nitrogen’s emission line. In our case:
NGC2336 / NGC2841λmin / 6597 / 6562,85
λmax / 6621 / 6582,99
The graphics showed us the emission peaks, of which we made a Gaussian fit, as showed in the figure.
GAUSSIAN FIT
DATA ELABORATION
Knowing that the wavelength of [NII] emission on earth is λ=6584 Ǻ, and that in the galactic centre the rotational velocity is null, we could find the recessional velocity of the entire galaxies using the red shift formula applied there, for example for the galaxy NGC2841: Then we divided the results among Hubble’s constant () finding the distances of the two
galaxies, expressed in Mpc, necessary to calculate conversion factors which allowed us to convert the pixels of the spectrum in parsec. The scale of the CCD used to obtain the spectrum is 1”/px so a px corresponds to an arc second and we didn’t need to transform pixels to arc seconds before converting them to parsec. To calculate the conversion factor we divided the distances of the bulge (in Pc) among 206265”: factor.
Those are the results:
Then we was able to obtain all the distances from the galactic centre expressed in Pc using the proportion: . Therefore we could calculate the velocity of every part of the galaxies in their mayor axis referred to centers just applying the red shift formula to the observed wavelength of [NII] emission and subtracting from them the recessional velocity of the centers. Then, we paired them to their distance from the centers and we represented them: Ngc2336
Ngc2841
CORRECTION OF VELOCITIES
But we had not consider that we was observing an inclined galaxy, and for this reason the translation of [N2] emission was minor than the real movement, because the velocity vector of the galactic disk is divided in two parts.
To calculate the angle of inclination we considered the triangle OAB, of the following picture, and we applied the rectangular triangle’s properties:
In the galaxies that we have studied the angles are:
To calculate the effective velocities we used rectangular triangle’s properties on the triangle having the observed velocity as cathetus and the effective velocity as hypotenuse, noticeable in the precedent picture:
CONSTRUCION OF MATHEMATICAL MODELS
After the calculation of real velocities, in the different points of the galaxies, we could proceed “building” the graphics of mathematical models that represents the rotation curves of the two galaxies. Using the mathematical models, not the observed velocities, we tried to reduce the observed reality, in which could be picked some experimental errors, to a “perfect” mathematic model, similar to the rotation curves simplifying their studies.
Observing the progress of the curves we can notice that in the zone near to the galactic centre there is a Newtonian movement, because velocity increases to the distance from the centre; instead in the more external zones, where we suppose the presence of Keplerian movement (because the galaxy isn’t a rigid object) the velocity approaches to be constant. This is a sign of the presence of dark matter, that we can’t see and that “pulls” even the more external zones of the galaxies.
NGC2236
NGC2841
The mathematic model that we used in this passage of our study has origins from literary, and we only modified the variable parameters (a, c₀, p) to adapt the mathematic function to the the obtained rotational curves:
MASS CALCULATION
We calculated the mass of the galaxies, using the velocity of the mathematic model and applying the Viriale’s theorem; this theorem affirms that in a close system the potential gravitational energy corresponds to the double of kinetics energy:
Observing the graphic of the masses we can see an exponential increase, because we calculated the mass of a relative radius including the masses relative to the more little radiuses, extending the spatial surface containing mass.
NGC2336
NGC2841
The mass values relative to galactic radiuses are:
Kinematics of two spiral galaxies Pag. 5
The sky as a laboratory / 2011Kinematics of two spiral galaxies Pag. 5