Merge 5d580c2 into 950f06f

.appveyor.yml

@@ -5,11 +5,11 @@ environment:
   MINICONDA: C:\\Miniconda36-x64
 
   matrix:
-    - TEST_FILES: "tests\\ --ignore=tests\\test_actionAngleTorus.py --ignore=tests\\test_snapshotpotential.py --ignore=tests\\test_qdf.py --ignore=tests\\test_pv2qdf.py --ignore=tests\\test_diskdf.py --ignore=tests\\test_orbit.py --ignore=tests\\test_streamdf.py --ignore=tests\\test_streamgapdf.py --ignore=tests\\test_evolveddiskdf.py --ignore=tests\\test_quantity.py --ignore=tests\\test_nemo.py --ignore=tests\\test_coords.py"
+    - TEST_FILES: "tests\\ --ignore=tests\\test_actionAngleTorus.py --ignore=tests\\test_snapshotpotential.py --ignore=tests\\test_qdf.py --ignore=tests\\test_pv2qdf.py --ignore=tests\\test_diskdf.py --ignore=tests\\test_orbit.py --ignore=tests\\test_orbits.py --ignore=tests\\test_streamdf.py --ignore=tests\\test_streamgapdf.py --ignore=tests\\test_evolveddiskdf.py --ignore=tests\\test_quantity.py --ignore=tests\\test_nemo.py --ignore=tests\\test_coords.py"
       ADDL_CONDA_PKGS: 
       COMPILE_NOOPENMP: 
 
-    - TEST_FILES: tests\test_orbit.py
+    - TEST_FILES: tests\test_orbit.py tests\test_orbits.py
       ADDL_CONDA_PKGS: astropy astroquery
       COMPILE_NOOPENMP: "--no-openmp"
 

.gitignore

@@ -7,6 +7,7 @@
 *.o
 *.so
 *.pyc
+*.pyd
 
 # Packages #
 ############

.travis.yml

@@ -11,17 +11,17 @@ env: #split tests
     - REQUIRES_ASTROQUERY=false
     - PYTHON_COVREPORTS_VERSION=3.7 # Version for which reports are uploaded
   matrix:
-    - TEST_FILES='tests/ --ignore=tests/test_qdf.py --ignore=tests/test_pv2qdf.py --ignore=tests/test_diskdf.py --ignore=tests/test_orbit.py --ignore=tests/test_streamdf.py --ignore=tests/test_streamgapdf.py --ignore=tests/test_evolveddiskdf.py --ignore=tests/test_quantity.py --ignore=tests/test_nemo.py --ignore=tests/test_coords.py --ignore=tests/test_jeans.py' REQUIRES_PYNBODY=true
+    - TEST_FILES='tests/ --ignore=tests/test_qdf.py --ignore=tests/test_pv2qdf.py --ignore=tests/test_diskdf.py --ignore=tests/test_orbit.py --ignore=tests/test_streamdf.py --ignore=tests/test_streamgapdf.py --ignore=tests/test_evolveddiskdf.py --ignore=tests/test_quantity.py --ignore=tests/test_nemo.py --ignore=tests/test_coords.py --ignore=tests/test_jeans.py --ignore=tests/test_orbits.py' REQUIRES_PYNBODY=true
     - TEST_FILES='tests/test_quantity.py tests/test_coords.py' REQUIRES_ASTROPY=true # needs to be separate for different config
-    - TEST_FILES='tests/test_orbit.py' REQUIRES_PYNBODY=true REQUIRES_ASTROPY=true REQUIRES_ASTROQUERY=true
+    - TEST_FILES='tests/test_orbit.py tests/test_orbits.py' REQUIRES_PYNBODY=true REQUIRES_ASTROPY=true REQUIRES_ASTROQUERY=true
     - TEST_FILES='tests/test_evolveddiskdf.py tests/test_jeans.py'
     - TEST_FILES='tests/test_diskdf.py'
     - TEST_FILES='tests/test_qdf.py tests/test_pv2qdf.py tests/test_streamgapdf.py'
     - TEST_FILES='tests/test_streamdf.py'
 matrix: # only run crucial tests for python 3.6
   include:
     - python: "3.6"
-      env: TEST_FILES='tests/test_orbit.py' REQUIRES_PYNBODY=true REQUIRES_ASTROPY=true REQUIRES_ASTROQUERY=true
+      env: TEST_FILES='tests/test_orbit.py tests/test_orbits.py' REQUIRES_PYNBODY=true REQUIRES_ASTROPY=true REQUIRES_ASTROQUERY=true
 addons:
   apt:
     packages:
@@ -75,21 +75,22 @@ install:
  - if [[ $TRAVIS_PYTHON_VERSION == $PYTHON_COVREPORTS_VERSION ]]; then easy_install --upgrade coveralls; fi
  - if $REQUIRES_PYNBODY; then pip install git+git://github.com/pynbody/pynbody.git; fi
  # clone my version of the torus code, don't do this for one test, to make sure the code installs without the torus code
- - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py' ]]; then git clone https://github.com/jobovy/Torus.git galpy/actionAngle/actionAngleTorus_c_ext/torus; fi
- - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py' ]]; then cd galpy/actionAngle/actionAngleTorus_c_ext/torus; fi
- - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py' ]]; then git checkout galpy; fi
- - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py' ]]; then cd -; fi
+ - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then git clone https://github.com/jobovy/Torus.git galpy/actionAngle/actionAngleTorus_c_ext/torus; fi
+ - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then cd galpy/actionAngle/actionAngleTorus_c_ext/torus; fi
+ - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then git checkout galpy; fi
+ - if [[ $TEST_FILES != 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then cd -; fi
  - if $REQUIRES_ASTROPY && [[ $TRAVIS_PYTHON_VERSION == 2.7 ]]; then conda install astropy --no-deps; fi
  - if $REQUIRES_ASTROPY && [[ $TRAVIS_PYTHON_VERSION != 2.7 ]]; then conda install astropy --no-deps; fi
  - if $REQUIRES_ASTROQUERY; then pip install astroquery; fi
- - python setup.py build_ext --coverage --single_ext --inplace
+# Compile w/o OpenMP for one, to make sure that works
+ - if [[ $TRAVIS_PYTHON_VERSION != $PYTHON_COVREPORTS_VERSION ]] && [[ $TEST_FILES == 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then python setup.py build_ext --coverage --single_ext --inplace --no-openmp; else python setup.py build_ext --coverage --single_ext --inplace; fi
  - python setup.py develop --single_ext
 # Following tests that we can start from an incomplete configuration file
- - if [[ $TEST_FILES == 'tests/test_evolveddiskdf.py' ]]; then echo -e '[normalization]' > $HOME/.galpyrc && echo -e 'ro = 8.' >> $HOME/.galpyrc && echo -e 'vo = 220.' >> $HOME/.galpyrc; fi
+ - if [[ $TEST_FILES == 'tests/test_evolveddiskdf.py tests/test_jeans.py' ]]; then echo -e '[normalization]' > $HOME/.galpyrc && echo -e 'ro = 8.' >> $HOME/.galpyrc && echo -e 'vo = 220.' >> $HOME/.galpyrc; fi
  - if [[ $TEST_FILES == 'tests/test_diskdf.py' ]]; then echo -e '[normalization]' > $HOME/.galpyrc && echo -e 'ro = 8.' >> $HOME/.galpyrc && echo -e '[astropy]' >> $HOME/.galpyrc && echo -e 'astropy-units = False' >> $HOME/.galpyrc && echo -e '[plot]' >> $HOME/.galpyrc && echo -e 'seaborn-bovy-defaults = True' >> $HOME/.galpyrc && echo -e '[warnings]' >> $HOME/.galpyrc && echo -e 'verbose = True' >> $HOME/.galpyrc; fi
 script:
 # only wait longer for orbit integrations
- - if [[ $TEST_FILES == 'tests/test_orbit.py' ]]; then travis_wait 40 pytest -v $TEST_FILES --cov galpy --cov-config .coveragerc_travis --disable-pytest-warnings; else pytest -v $TEST_FILES --cov galpy --cov-config .coveragerc_travis --disable-pytest-warnings; fi
+ - if [[ $TEST_FILES == 'tests/test_orbit.py tests/test_orbits.py' ]]; then travis_wait 40 pytest -v $TEST_FILES --cov galpy --cov-config .coveragerc_travis --disable-pytest-warnings; else pytest -v $TEST_FILES --cov galpy --cov-config .coveragerc_travis --disable-pytest-warnings; fi
 after_success:
  # Generate lcov output 
  - if [[ $TRAVIS_PYTHON_VERSION == $PYTHON_COVREPORTS_VERSION ]]; then lcov --capture --base-directory . --directory build/temp.linux-x86_64-3.7/galpy/ --no-external --output-file coverage_full.info; fi

HISTORY.txt

@@ -1,6 +1,11 @@
 v1.5 (????-??-??)
 =================
 
+- Added support for holding multiple objects in an Orbit instance,
+  with efficient handling of multiple objects and parallelized
+  integration and action-angle-related functions. Orbit instances can
+  now have arbitrary shapes. Full re-write of Orbit class (PR #384).
+
 - Added support for 1D orbit integration in C (PR #354).
 
 - Added potential.toVerticalPotential to convert any 3D potential to a

doc/source/actionAngle.rst

@@ -10,6 +10,9 @@ preferred method for accessing them is through the routines in this
 module. There is also some support for accessing the actionAngle
 routines as methods of the ``Orbit`` class.
 
+.. TIP::
+   If you want to quickly and easily compute actions, angles, or frequencies using the Staeckel approximation, using the ``Orbit`` interface as described in :ref:`this section <aaorbit>` is recommended. Especially if you are starting from observed coordinates, as ``Orbit`` instances can easily be initialized using these.
+
 Since v1.2, galpy can also compute positions and velocities
 corresponding to a given set of actions and angles for axisymmetric
 potentials using the TorusMapper code of `Binney & McMillan (2016)
@@ -830,14 +833,16 @@ action-angle API page for computing Hessians and Jacobians of the
 transformation between action-angle and configuration space
 coordinates.
 
+.. _aaorbit:
+
 Accessing action-angle coordinates for Orbit instances
-----------------------------------------------------------
+------------------------------------------------------
 
-While the recommended way to access the actionAngle routines is
+While the most flexible way to access the actionAngle routines is
 through the methods in the ``galpy.actionAngle`` modules, action-angle
-coordinates can also be calculated for ``galpy.orbit.Orbit``
-instances. This is illustrated here briefly. We initialize an Orbit
-instance
+coordinates can also be calculated for ``galpy.orbit.Orbit`` instances
+and this is often more convenient. This is illustrated here
+briefly. We initialize an Orbit instance
 
 >>> from galpy.orbit import Orbit
 >>> from galpy.potential import MWPotential2014
@@ -847,52 +852,70 @@ and we can then calculate the actions (default is to use the staeckel
 approximation with an automatically-estimated delta parameter, but
 this can be adjusted)
 
->>> o.jr(MWPotential2014), o.jp(MWPotential2014), o.jz(MWPotential2014)
+>>> o.jr(pot=MWPotential2014), o.jp(pot=MWPotential2014), o.jz(pot=MWPotential2014)
 # (0.018194068808944613,1.1,0.01540155584446606)
 
 ``o.jp`` here gives the azimuthal action (which is the *z* component
 of the angular momentum for axisymmetric potentials). We can also use
 the other methods described above or adjust the parameters of the
 approximation (see above):
 
->>> o.jr(MWPotential2014,type='staeckel',delta=0.4), o.jp(MWPotential2014,type='staeckel',delta=0.4), o.jz(MWPotential2014,type='staeckel',delta=0.4)
+>>> o.jr(pot=MWPotential2014,type='staeckel',delta=0.4), o.jp(pot=MWPotential2014,type='staeckel',delta=0.4), o.jz(pot=MWPotential2014,type='staeckel',delta=0.4)
 # (0.019221672966336707, 1.1, 0.015276825017286827)
->>> o.jr(MWPotential2014,type='adiabatic'), o.jp(MWPotential2014,type='adiabatic'), o.jz(MWPotential2014,type='adiabatic')
+>>> o.jr(pot=MWPotential2014,type='adiabatic'), o.jp(pot=MWPotential2014,type='adiabatic'), o.jz(pot=MWPotential2014,type='adiabatic')
 # (0.016856430059017123, 1.1, 0.015897730620467752)
->>> o.jr(MWPotential2014,type='isochroneApprox',b=0.8), o.jp(MWPotential2014,type='isochroneApprox',b=0.8), o.jz(MWPotential2014,type='isochroneApprox',b=0.8)
+>>> o.jr(pot=MWPotential2014,type='isochroneApprox',b=0.8), o.jp(pot=MWPotential2014,type='isochroneApprox',b=0.8), o.jz(pot=MWPotential2014,type='isochroneApprox',b=0.8)
 # (0.019066091295488922, 1.1, 0.015280492319332751)
 
 These two methods give very precise actions for this orbit (both are
 converged to about 1%) and they agree very well
 
->>> (o.jr(MWPotential2014,type='staeckel',delta=0.4)-o.jr(MWPotential2014,type='isochroneApprox',b=0.8))/o.jr(MWPotential2014,type='isochroneApprox',b=0.8)
+>>> (o.jr(pot=MWPotential2014,type='staeckel',delta=0.4)-o.jr(pot=MWPotential2014,type='isochroneApprox',b=0.8))/o.jr(pot=MWPotential2014,type='isochroneApprox',b=0.8)
 # 0.00816012408818143
->>> (o.jz(MWPotential2014,type='staeckel',delta=0.4)-o.jz(MWPotential2014,type='isochroneApprox',b=0.8))/o.jz(MWPotential2014,type='isochroneApprox',b=0.8)
+>>> (o.jz(pot=MWPotential2014,type='staeckel',delta=0.4)-o.jz(pot=MWPotential2014,type='isochroneApprox',b=0.8))/o.jz(pot=MWPotential2014,type='isochroneApprox',b=0.8)
 # 0.00023999894566772273
 
-.. WARNING:: Once an action, frequency, or angle is calculated for a given type of calculation (e.g., staeckel), the parameters for that type are fixed in the Orbit instance. Call o.resetaA() to reset the action-angle instance used when using different parameters (i.e., different ``delta=`` for staeckel or different ``b=`` for isochroneApprox.
-
 We can also calculate the frequencies and the angles. This requires
 using the Staeckel or Isochrone approximations, because frequencies
 and angles are currently not supported for the adiabatic
 approximation. For example, the radial frequency
 
->>> o.Or(MWPotential2014,type='staeckel',delta=0.4)
+>>> o.Or(pot=MWPotential2014,type='staeckel',delta=0.4)
 # 1.1131779637307115
->>> o.Or(MWPotential2014,type='isochroneApprox',b=0.8)
+>>> o.Or(pot=MWPotential2014,type='isochroneApprox',b=0.8)
 # 1.1134635974560649
 
 and the radial angle
 
->>> o.wr(MWPotential2014,type='staeckel',delta=0.4)
+>>> o.wr(pot=MWPotential2014,type='staeckel',delta=0.4)
 # 0.37758086786371969
->>> o.wr(MWPotential2014,type='isochroneApprox',b=0.8)
+>>> o.wr(pot=MWPotential2014,type='isochroneApprox',b=0.8)
 # 0.38159809018175395
 
 which again agree to 1%. We can also calculate the other frequencies,
 angles, as well as periods using the functions ``o.Op``, ``o.oz``,
 ``o.wp``, ``o.wz``, ``o.Tr``, ``o.Tp``, ``o.Tz``.
 
+All of the functions above also work for ``Orbit`` instances that
+contain multiple objects. This is particularly convenient if you have
+data in observed coordinates (e.g., RA, Dec, etc.), for example,
+
+>>> numpy.random.seed(1)
+>>> nrand= 30
+>>> ras= numpy.random.uniform(size=nrand)*360.*u.deg
+>>> decs= 90.*(2.*numpy.random.uniform(size=nrand)-1.)*u.deg
+>>> dists= numpy.random.uniform(size=nrand)*10.*u.kpc
+>>> pmras= 2.*(2.*numpy.random.uniform(size=nrand)-1.)*u.mas/u.yr
+>>> pmdecs= 2.*(2.*numpy.random.uniform(size=nrand)-1.)*u.mas/u.yr
+>>> vloss= 200.*(2.*numpy.random.uniform(size=nrand)-1.)*u.km/u.s
+>>> co= SkyCoord(ra=ras,dec=decs,distance=dists,
+                 pm_ra_cosdec=pmras,pm_dec=pmdecs,
+                 radial_velocity=vloss,
+                 frame='icrs')
+>>> orbits= Orbit(co)
+>>> orbits.jr(pot=MWPotential2014)
+# [2363.7957, 360.12445, 690.32238, 1046.2924, 132.9572, 86.989812, 272.06487, 360.73566, 55.568238, 698.18447, 24.783574, 21.889352, 16.148216, 3870.4286, 743.63456, 317.66551, 325.93816, 183.86429, 56.087796, 180.42838, 1121.8019, 8700.8335, 977.8525, 7.569396, 8.2847477, 210.72127, 160.9785, 680.63864, 1093.7413, 87.629873]kmkpcs
+
 Example: Evidence for a Lindblad resonance in the Solar neighborhood
 ---------------------------------------------------------------------
 

doc/source/basic_df.rst

@@ -529,13 +529,15 @@ time-coordinates).
 
 We initialize orbits on a grid in velocity space and integrate them
 
->>> ins=[[Orbit([1.,-0.7+1.4/100*jj,1.-0.6+1.2/100*ii,0.]) for jj in range(101)] for ii in range(101)]
->>> int=[[o.integrate(ts,[lp,dp]) for o in j] for j in ins]
+>>> ins= Orbit(numpy.array([[[1.,-0.7+1.4/100*jj,1.-0.6+1.2/100*ii,0.] for jj in range(101)] for ii in range(101)]))
+>>> ins.integrate(ts,[lp,dp])
 
 We can then evaluate the weight of these orbits by assuming that the
 disk was in a steady-state before bar-formation with a Dehnen
 distribution function. We evaluate the Dehnen distribution function at
-``dp.tform()`` for each of the orbits
+``dp.tform()`` for each of the orbits (evaluating the distribution
+function only works for an Orbit with a single object, so we need to
+unpack the Orbit instance that contains all orbits)
 
 >>> dfc= dehnendf(beta=0.,correct=True)
 >>> out= [[dfc(o(dp.tform())) for o in j] for j in ins]

doc/source/examples/dierickx_eccentricities.py

@@ -26,39 +26,15 @@ def calc_eccentricity(args, options):
     dierickx = ascii.read(table, readme=readme)
     vxvv = np.dstack([dierickx['RAdeg'], dierickx['DEdeg'], dierickx['Dist']/1e3, dierickx['pmRA'], dierickx['pmDE'], dierickx['HRV']])[0]
     ro, vo, zo = 8., 220., 0.025
-    ra, dec= vxvv[:,0], vxvv[:,1]
-    lb= bovy_coords.radec_to_lb(ra,dec,degree=True)
-    pmra, pmdec= vxvv[:,3], vxvv[:,4]
-    pmllpmbb= bovy_coords.pmrapmdec_to_pmllpmbb(pmra,pmdec,ra,dec,degree=True)
-    d, vlos= vxvv[:,2], vxvv[:,5]
-    rectgal= bovy_coords.sphergal_to_rectgal(lb[:,0],lb[:,1],d,vlos,pmllpmbb[:,0], pmllpmbb[:,1],degree=True)
-    vsolar= np.array([-10.1,4.0,6.7])
-    vsun= np.array([0.,1.,0.,])+vsolar/vo
-    X = rectgal[:,0]/ro
-    Y = rectgal[:,1]/ro
-    Z = rectgal[:,2]/ro
-    vx = rectgal[:,3]/vo
-    vy = rectgal[:,4]/vo
-    vz = rectgal[:,5]/vo
-    vsun= np.array([0.,1.,0.,])+vsolar/vo
-    Rphiz= bovy_coords.XYZ_to_galcencyl(X,Y,Z,Zsun=zo/ro)
-    vRvTvz= bovy_coords.vxvyvz_to_galcencyl(vx,vy,vz,Rphiz[:,0],Rphiz[:,1],Rphiz[:,2],vsun=vsun,Xsun=1.,Zsun=zo/ro,galcen=True)
+    orbits= Orbit(vxvv,radec=True,ro=ro,vo=vo,zo=zo,solarmotion='hogg')
     #do the integration and individual analytic estimate for each object
-    ts= np.linspace(0.,20.,10000)
     lp= LogarithmicHaloPotential(normalize=1.)
-    e_ana = numpy.zeros(len(vxvv))
-    e_int = numpy.zeros(len(vxvv))
-    print('Performing orbit integration and analytic parameter estimates for Dierickx et al. sample...')
-    for i in tqdm(range(len(vxvv))):
-        try:
-            orbit = Orbit(vxvv[i], radec=True, vo=220., ro=8.)
-            e_ana[i] = orbit.e(analytic=True, pot=lp, c=True)
-        except UnboundError:
-            e_ana[i] = np.nan
-        orbit.integrate(ts, lp)
-        e_int[i] = orbit.e(analytic=False)
+    e_ana= orbits.e(analytic=True,pot=lp,delta=1e-6)
+    ts= np.linspace(0.,20.,10000)
+    orbits.integrate(ts,lp)
+    e_int= orbits.e()
+    # Now plot everything
     fig = plt.figure()
-    fig.set_size_inches(1.5*columnwidth, 1.5*columnwidth)
     plt.scatter(e_int, e_ana,  s=1, color='Black', lw=0.)
     plt.xlabel(r'$\mathrm{galpy\ integrated}\ e$')
     plt.ylabel(r'$\mathrm{galpy\ analytic}\ e$')
@@ -67,14 +43,12 @@ def calc_eccentricity(args, options):
     fig.tight_layout()
     plt.savefig(os.path.join(args[0],'dierickx-integratedeanalytice.png'), format='png', dpi=200)
     fig = plt.figure()
-    fig.set_size_inches(1.5*columnwidth, 1.5*columnwidth)
     plt.hist(e_int, bins=30)
     plt.xlim(0.,1.)
     plt.xlabel(r'$\mathrm{galpy}\ e$')
     fig.tight_layout()
     plt.savefig(os.path.join(args[0], 'dierickx-integratedehist.png'), format='png', dpi=200)
     fig = plt.figure()
-    fig.set_size_inches(1.5*columnwidth, 1.5*columnwidth)
     plt.scatter(dierickx['e'], e_int,  s=1, color='Black', lw=0.)
     plt.xlabel(r'$\mathrm{Dierickx\ et\ al.}\ e$')
     plt.ylabel(r'$\mathrm{galpy\ integrated}\ e$')
@@ -83,7 +57,6 @@ def calc_eccentricity(args, options):
     fig.tight_layout()
     plt.savefig(os.path.join(args[0],'dierickx-integratedee.png'), format='png', dpi=200)
     fig = plt.figure()
-    fig.set_size_inches(1.5*columnwidth, 1.5*columnwidth)
     plt.scatter(dierickx['e'], e_ana,  s=1, color='Black', lw=0.)
     plt.xlabel(r'$\mathrm{Dierickx\ et\ al.}\ e$')
     plt.ylabel(r'$\mathrm{galpy\ estimated}\ e$')
@@ -188,4 +161,4 @@ def get_options():
     parser = get_options()
     options, args= parser.parse_args()
     get_table(args,options)
-    calc_eccentricity(args, options)
+    calc_eccentricity(args, options)

doc/source/getting_started.rst

@@ -307,7 +307,7 @@ 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])
+>>> o= Orbit([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
@@ -319,7 +319,7 @@ azimuth. We then integrate the orbit for a set of times ``ts``
 >>> o.integrate(ts,mp,method='odeint')
 
 .. TIP::
-   Like for the Miyamoto-Nagai example in the section above, the Orbit and integration times can also be specified in physical units, e.g., ``o= Orbit(vxvv=[8.*units.kpc,22.*units.km/units.s,242.*units.km/units.s.0.*units.pc,20.*units.km/s])`` and ``ts= numpy.linspace(0.,10.,10000)*units.Gyr``
+   Like for the Miyamoto-Nagai example in the section above, the Orbit and integration times can also be specified in physical units, e.g., ``o= Orbit([8.*units.kpc,22.*units.km/units.s,242.*units.km/units.s.0.*units.pc,20.*units.km/s])`` and ``ts= numpy.linspace(0.,10.,10000)*units.Gyr``
 
 Now we plot the resulting orbit as