## 3.1 Radial motion of whole bodies ref: [[Part 1 Astrometry measuring distances]] #### 3.1.1 The Doppler effect The radial velocity can be calculated (for non-relativistic speeds): *v$_r$ = $\dfrac{c (\lambda_{obs}- \lambda_{em})}{\lambda_{em}}$ #### 3.1.2 Measuring radial velocities check interactive graph: https://learn2.open.ac.uk/mod/oucontent/view.php?id=2158707&section=3.1.2 python : ~~~ python def calculate_missing_parameter(): print("Enter two out of the three parameters: observed wavelength (nm), emitted wavelength (nm), radial velocity (m/s).") print("For the parameter you want to calculate, enter 'unknown'.") lambda_obs_input = input("Enter observed wavelength (nm) or 'unknown': ").strip().lower() lambda_emit_input = input("Enter emitted wavelength (nm) or 'unknown': ").strip().lower() velocity_input = input("Enter radial velocity (km/s) or 'unknown': ").strip().lower() speed_of_light = 3e5 # Speed of light in kilometers per second try: if 'unknown' not in [lambda_obs_input, lambda_emit_input, velocity_input]: print("Please ensure one and only one parameter is set to 'unknown'.") return if lambda_obs_input == 'unknown': lambda_emit = float(lambda_emit_input) velocity = float(velocity_input) lambda_obs = lambda_emit * (1 + velocity / speed_of_light) print(f"The calculated observed wavelength is {lambda_obs} nm.") elif lambda_emit_input == 'unknown': lambda_obs = float(lambda_obs_input) velocity = float(velocity_input) lambda_emit = lambda_obs / (1 + velocity / speed_of_light) print(f"The calculated emitted wavelength is {lambda_emit} nm.") elif velocity_input == 'unknown': lambda_obs = float(lambda_obs_input) lambda_emit = float(lambda_emit_input) velocity = speed_of_light * ((lambda_obs - lambda_emit) / lambda_emit) print(f"The calculated radial velocity is {velocity} km/s.") except ValueError: print("Invalid input. Please enter numerical values for known parameters and 'unknown' for the unknown parameter.") calculate_missing_parameter() ~~~ #### 3.1.3 Space velocities *v = $\sqrt{v_{t}^2 + v_{r}^2}$ #### 3.1.4 Redshift *z = $\dfrac{(\lambda_{obs}- \lambda_{em})}{\lambda_{em}}$ = $\dfrac{\lambda_{obs}}{\lambda_{em}} -1$* So *v$_r$ = zc* Within the local group the galaxies might be red or blue shifted but outside the local group the galaxies are all red shifted eg moving away. This due the expansion of space itself. This redshift is called [[../Glossary/cosmological redshift]]. ## 3.2 Motion of two bodies around a common centre of mass #### 3.2.1 Orbits *Barycentre* is the center of mass of a system Every orbit is an elliptical orbit which characterised by its [[../Glossary/semi-major axis]] *a* and semiminor axis *b*. In an ellipse, is equivalent to the average distance between and in a binary system (equal to for a circle) sits at one focus of the ellipse, and orbits it on the boundary of the ellipse. The foci are located at a distance *c* from the centre where *c = $\sqrt{a^2-b^2}$ . ![[../Assets/Pasted image 20231114150918.png|450]] The degree of [[../Glossary/eccentricity]] of an ellips is defined as half the distance between the two foci, divided by *a* . The [[../Glossary/inclination]]of a binary orbit is angle between the plane of the orbit and the plane of the sky. #### 3.2.2. Spectroscopic binary stars The individual spectra of both stars will effectively be added together and, if both are bright enough, we will see spectral lines from both stars (a [[../Glossary/double lined spectroscopic binary]]), superimposed on a combined continuum spectrum. If only one star’s lines are detectable, it would be a [[../Glossary/single-lined spectroscopic binary]]. #### 3.2.3. Using spectroscopic binaries Kepler's third law: *P$_{orb}^2$ = $\dfrac{4\pi^2 a^3}{G(M+m)}$* (G = 6.67 x 10$^{-11}$ Nm$^2$kg$^{-2}$ ) or normalised: (P$_{orb}$ / y)$^2$ = $\dfrac {a^3}{M+m}$ ( where a is in AU and the masses are solar masses) So if we know the period and the semi-major axis we can calculate the total mass of the binary system. The period can be derived from a light curve or from spectroscopic [[../Glossary/radial velocity curve]]. From the radial velocity curve we can also derive the semi-major axis: *v$_{RM}$ = $\dfrac{2\pi d_{M}}{P_{{orb}}}$* and *v$_{Rm}$ = $\dfrac{2\pi d_{m}}{P_{{orb}}}$* also: $\dfrac{M}{m}$ = $\dfrac{d_m}{d_M}$ so we have M+m and M/m so we can determine the individual stellar masses. ## 3.3 Motion of many bodies about a common centre of mass #### 3.3.1 Rotation curvesn curves [[../Glossary/Keplerian rotation]] is described by the equation $v = \sqrt{\dfrac{GM}{R}}$ ![[../Assets/Pasted image 20231127102932.png]] #### 3.3.2 Galactic rotation curves The most recent observations for the outer Galaxy (shown in blue), based on the proper motions of Cepheids observed with Gaia, show a flattening of the rotation curve out to about 20 kpc, extending beyond the visible stellar disc of the Milky Way. ![[../Assets/Pasted image 20231128131036.png|450]] ## 3.4 Motion of components within a body #### 3.4.1. Motions in a gas The [[../Glossary/thermal motion]] of a gas is a random movement of atoms in a gas associated with the temperature and the mass of the atoms, producing a velocity dispersion: $\Delta$v = $\sqrt{\dfrac{2kT}{m}}$ {k = 1.38 x 10$^{-23}$ JK$^{-1}$} This random movement gives rise to a [[../Glossary/Doppler broadening]] of the spectral lines. the velocity dispersion is related to the Doppler broadening by : $\dfrac{\delta \lambda}{\lambda}$ ~ $\dfrac{\delta v}{c}$ Another significant contributor to Doppler broadening is [[../Glossary/bulk motion]]. it broadens all the spectral lines evenly. #### 3.4.2 Motions in extended objects To measure the the speeds / rotation curves for a galaxy, the 21 cm line is measured at different spots of the galaxy. However for distant unresolved galaxies the 21 cm line for the whole galaxy is measured. This gives us the max rotation speed of the galaxy which corresponds to half the total line width. There is a relation between the luminosity of a galaxy and the max rotation speed, [[../Glossary/The Tully-Fisher relation]]: *L $\propto$ v$_{max}^4$* Similarly there is a relation between the velocity dispersion of the stars in [[../Glossary/elliptical galaxies]] and the l[[../Glossary/luminosity]]. [[../Glossary/The Faber-Jackson relation]]. *L $\propto$ $\Delta v^4$* #### 3.4.3 Motion in a star. [[../Glossary/bulk motion]] within a star: the subject of [[../Glossary/asteroseismology.]] In the Sun surface rises and fall with app 500 ms$^{-1}$. Lightcurves can reveal these oscillations. ## reference/links [[../Assets/LaTex]] [[Astronomy]] [[../constants & formula's]]