Introduction ============= The most basic features of galpy are its ability to display rotation curves and perform orbit integration for arbitrary combinations of potentials. This section introduce the most basic features of ``galpy.potential`` and ``galpy.orbit``. .. _rotcurves: Rotation curves --------------- The following code example shows how to initialize a Miyamoto-Nagai disk potential and plot its rotation curve >>> from galpy.potential import MiyamotoNagaiPotential >>> mp= MiyamotoNagaiPotential(a=0.5,b=0.0375,normalize=1.) >>> mp.plotRotcurve(Rrange=[0.01,10.],grid=1001) The ``normalize=1.`` option normalizes the potential such that the radial force is a fraction ``normalize=1.`` of the radial force necessary to make the circular velocity 1 at R=1. Similarly we can initialize other potentials and plot the combined rotation curve >>> from galpy.potential import NFWPotential, HernquistPotential >>> mp= MiyamotoNagaiPotential(a=0.5,b=0.0375,normalize=.6) >>> np= NFWPotential(a=4.5,normalize=.35) >>> hp= HernquistPotential(a=0.6/8,normalize=0.05) >>> from galpy.potential import plotRotcurve >>> plotRotcurve([hp,mp,np],Rrange=[0.01,10.],grid=1001,yrange=[0.,1.2]) Note that the ``normalize`` values add up to 1. such that the circular velocity will be 1 at R=1. The resulting rotation curve is approximately flat. To show the rotation curves of the three components do >>> mp.plotRotcurve(Rrange=[0.01,10.],grid=1001,overplot=True) >>> hp.plotRotcurve(Rrange=[0.01,10.],grid=1001,overplot=True) >>> np.plotRotcurve(Rrange=[0.01,10.],grid=1001,overplot=True) You'll see the following .. image:: images/rotcurve.png As a shortcut the ``[hp,mp,np]`` Milky-Way-like potential is defined as >>> from galpy.potential import MWPotential This is *not* the recommended Milky-Way-like potential in ``galpy``. The (currently) recommended Milky-Way-like potential is ``MWPotential2014``: >>> from galpy.potential import MWPotential2014 ``MWPotential2014`` has a more realistic bulge model and is actually fit to various dynamical constraints on the Milky Way (see :ref:`here ` and the ``galpy`` paper). Units in galpy -------------- .. _units: Above we normalized the potentials such that they give a circular velocity of 1 at R=1. These are the standard galpy units (sometimes referred to as *natural units* in the documentation). galpy will work most robustly when using these natural units. When using galpy to model a real galaxy with, say, a circular velocity of 220 km/s at R=8 kpc, all of the velocities should be scaled as v= V/[220 km/s] and all of the positions should be scaled as x = X/[8 kpc] when using galpy's natural units. For convenience, a utility module ``bovy_conversion`` is included in galpy that helps in converting between physical units and natural units for various quantities. For example, in natural units the orbital time of a circular orbit at R=1 is :math:`2\pi`; in physical units this corresponds to >>> from galpy.util import bovy_conversion >>> print 2.*numpy.pi*bovy_conversion.time_in_Gyr(220.,8.) 0.223405444283 or about 223 Myr. We can also express forces in various physical units. For example, for the Milky-Way-like potential defined in galpy, we have that the vertical force at 1.1 kpc is >>> from galpy.potential import MWPotential2014, evaluatezforces >>> -evaluatezforces(1.,1.1/8.,MWPotential2014)*bovy_conversion.force_in_pcMyr2(220.,8.) 2.0259181908629933 which we can also express as an equivalent surface-density by dividing by :math:`2\pi G` >>> -evaluatezforces(1.,1.1/8.,MWPotential2014)*bovy_conversion.force_in_2piGmsolpc2(220.,8.) 71.658016957792356 Because the vertical force at the solar circle in the Milky Way at 1.1 kpc above the plane is approximately :math:`70\,(2\pi G\, M_\odot\,\mathrm{pc}^{-2})` (e.g., `2013arXiv1309.0809B `_), this shows that our Milky-Way-like potential has a realistic disk (at least in this respect). ``bovy_conversion`` further has functions to convert densities, masses, surface densities, and frequencies to physical units (actions are considered to be too obvious to be included); see :ref:`here ` for a full list. As a final example, the local dark matter density in the Milky-Way-like potential is given by >>> MWPotential2014[2].dens(1.,0.)*bovy_conversion.dens_in_msolpc3(220.,8.) 0.0075419566970079373 or >>> MWPotential2014[2].dens(1.,0.)*bovy_conversion.dens_in_gevcc(220.,8.) 0.28643101789044584 or about :math:`0.0075\,M_\odot\,\mathrm{pc}^{-3} \approx 0.3\,\mathrm{GeV\,cm}^{-3}`, in line with current measurements (e.g., `2012ApJ...756...89B `_). When ``galpy`` Orbits are initialized using a distance scale ``ro=`` and a velocity scale ``vo=`` output quantities returned and plotted in physical coordinates. Specifically, positions are are returned in units of kpc, velocities in km/s, energies and the Jacobi integral in (km/s)^2, the angular momentum o.L() and actions in km/s kpc, frequencies in 1/Gyr, and times and periods in Gyr. Orbit integration ----------------- We can also integrate orbits in all galpy potentials. Going back to a simple Miyamoto-Nagai potential, we initialize an orbit as follows >>> from galpy.orbit import Orbit >>> mp= MiyamotoNagaiPotential(a=0.5,b=0.0375,amp=1.,normalize=1.) >>> o= Orbit(vxvv=[1.,0.1,1.1,0.,0.1]) Since we gave ``Orbit()`` a five-dimensional initial condition ``[R,vR,vT,z,vz]``, we assume we are dealing with a three-dimensional axisymmetric potential in which we do not wish to track the azimuth. We then integrate the orbit for a set of times ``ts`` >>> import numpy >>> ts= numpy.linspace(0,100,10000) >>> o.integrate(ts,mp,nethod='odeint') Now we plot the resulting orbit as >>> o.plot() Which gives .. image:: images/mp-orbit-integration.png The integrator used is not symplectic, so the energy error grows with time, but is small nonetheless >>> o.plotE(normed=True) .. image:: images/mp-orbit-E.png When we use a symplectic leapfrog integrator, we see that the energy error remains constant >>> o.integrate(ts,mp,method='leapfrog') >>> o.plotE(xlabel=r'$t$',ylabel=r'$E(t)/E(0)$') .. image:: images/mp-orbit-Esymp.png Because stars have typically only orbited the center of their galaxy tens of times, using symplectic integrators is mostly unnecessary (compared to planetary systems which orbits millions or billions of times). galpy contains fast integrators written in C, which can be accessed through the ``method=`` keyword (e.g., ``integrate(...,method='dopr54_c')`` is a fast high-order Dormand-Prince method). When we integrate for much longer we see how the orbit fills up a torus (this could take a minute) >>> ts= numpy.linspace(0,1000,10000) >>> o.integrate(ts,mp,method='odeint') >>> o.plot() .. image:: images/mp-long-orbit-integration.png As before, we can also integrate orbits in combinations of potentials. Assuming ``mp, np,`` and ``hp`` were defined as above, we can >>> ts= numpy.linspace(0,100,10000) >>> o.integrate(ts,[mp,hp,np]) >>> o.plot() .. image:: images/mphpnp-orbit-integration.png Energy is again approximately conserved >>> o.plotE(xlabel=r'$t$',ylabel=r'$E(t)/E(0)$') .. image:: images/mphpnp-orbit-E.png Escape velocity curves ---------------------- Just like we can plot the rotation curve for a potential or a combination of potentials, we can plot the escape velocity curve. For example, the escape velocity curve for the Miyamoto-Nagai disk defined above >>> mp.plotEscapecurve(Rrange=[0.01,10.],grid=1001) .. image:: images/esc-miyamoto.png or of the combination of potentials defined above >>> from galpy.potential import plotEscapecurve >>> plotEscapecurve([mp,hp,np],Rrange=[0.01,10.],grid=1001) .. image:: images/esc-comb.png For the Milky-Way-like potential ``MWPotential2014``, the escape-velocity curve is >>> plotEscapecurve(MWPotential2014,Rrange=[0.01,10.],grid=1001) .. image:: images/esc-mw14.png At the solar radius, the escape velocity is >>> from galpy.potential import vesc >>> vesc(MWPotential2014,1.) 2.3316389848832784 Or, for a local circular velocity of 220 km/s >>> vesc(MWPotential2014,1.)*220. 512.96057667432126 similar to direct measurements of this (e.g., `2007MNRAS.379..755S `_ and `2014A%26A...562A..91P `_).