Guiding Center Radius Research Articles

North–South asymmetries in the Galactic thin disc associated with the vertical phase spiral as seen using LAMOST-Gaia stars

ABSTRACT We select 1052 469 (754 635) thin disc stars from Gaia eDR3 and LAMOST DR7 in the range of Galactocentric radius R (guiding centre radius Rg) from 8 to 11 kpc to investigate the asymmetries between the North and South of the disc mid-plane. More specifically, we analyse the vertical velocity dispersion profiles ($\sigma _{v_{z}}(z$)) in different bins of R (Rg) and [Fe/H]. We find troughs in the profiles of $\sigma _{v_{z}}(z)$ located in both the North (z ∼ 0.7 kpc) and South (z ∼ −0.5 kpc) of the disc at all radial and chemical bins studied. The difference between the Northern and Southern vertical velocity dispersion profiles ($\Delta \sigma _{v_{z}}(|z|)$) shows a shift between curves of different R and Rg. A similar shift exists in these North–South (NS) asymmetry profiles further divided into different [Fe/H] ranges. The sample binned with Rg more clearly displays the features in the velocity dispersion profiles. The shift in the peaks of the $\Delta \sigma _{v_{z}}$ profiles and the variation in the phase spiral shape binned by metallicity indicate the variation of the vertical potential profiles and the radial metallicity gradient. The wave-like signal in NS asymmetry of $\sigma _{v_{z}}(z)$ largely originates from phase spiral; while the NS asymmetry profiles of [Fe/H] only display a weak wave-like feature near solar radius. We perform a test particle simulation to qualitatively reproduce the observed results. A quantitative explanation of the NS asymmetry in the metallicity profile needs careful consideration of the spiral shape and the perturbation model, and we leave this for future work.

Measuring the Milky Way Vertical Potential with the Phase Snail in a Model-independent Way

The vertical phase-space spiral (snail) is a direct sign of disequilibrium of the Milky Way’s disk. Nevertheless, the wrapping of the phase snail contains information on the vertical potential. We propose a novel method to measure the vertical potential utilizing the intersections between the snail and z/V z axes, for which we know the maximum vertical heights (Z max) or velocities ( Vz,max ). Using a refined linear interpolation method, we directly obtain (Zmax,12Vz,max2) for these snail intersections to constrain the vertical potential profile empirically. Our method is model-independent, since no assumptions about the snail shape or the vertical potential have been made. Although the snail binned by the guiding center radius (R g ) is more prominent, it traces a vertical potential shallower than that of the snail binned by the same galactocentric radius (R). We apply an empirical method to correct this difference. We measure the snail intersections in several R g bins within 7.5 < R g < 11.0 kpc for Gaia DR3 and apply the interpolation method to deduce the potential values at several vertical heights. The potential at the snail intersections, as well as the following mass modeling, is consistent with the popular Milky Way potentials in the literature. For the R g -binned phase snail in the solar neighborhood, the mass modeling indicates a local dark matter density of ρ dm = 0.0150 ± 0.0031 M ⊙ pc−3, consistent with previous works. Our method could be applied to larger radial ranges in future works to provide independent and stronger constraints on the Milky Way’s potential.

The Open Cluster Chemical Abundances and Mapping Survey. VI. Galactic Chemical Gradient Analysis from APOGEE DR17

The goal of the Open Cluster Chemical Abundances and Mapping (OCCAM) survey is to constrain key Galactic dynamic and chemical evolution parameters by the construction and analysis of a large, comprehensive, uniform data set of infrared spectra for stars in hundreds of open clusters. This sixth contribution from the OCCAM survey presents analysis of SDSS/APOGEE Data Release 17 (DR17) results for a sample of stars in 150 open clusters, 94 of which we designate to be “high-quality” based on the appearance of their color–magnitude diagram. We find the APOGEE DR17-derived [Fe/H] values to be in good agreement with those from previous high-resolution spectroscopic open cluster abundance studies. Using a subset of the high-quality sample, the Galactic abundance gradients were measured for 16 chemical elements, including [Fe/H], for both Galactocentric radius (R GC) and guiding center radius (R guide). We find an overall Galactic [Fe/H] versus R GC gradient of −0.073 ± 0.002 dex kpc−1 over the range of 6 > R GC < 11.5 kpc, and a similar gradient is found for [Fe/H] versus R guide. Significant Galactic abundance gradients are also noted for O, Mg, S, Ca, Mn, Na, Al, K, and Ce. Our large sample additionally allows us to explore the evolution of the gradients in four age bins for the remaining 15 elements.

Development of a Magnetic Cusp Gun for a 1-THz Harmonic Gyrotron

High-quality axis-encircling electron beam is essential to drive a high-harmonic gyrotron. A magnetic cusp gun that can generate a low-spread axis-encircling electron beam at an extremely high magnetic-compression ratio was developed for a 1-THz fourth-harmonic gyrotron toward continuous-wave operation. The optimized gun could generate an axis-encircling electron beam with 80-kV voltage, 1-A current, and a nearly constant pitch factor of about 1.5 in a wide range of magnetic field strength. The transverse velocity spread was less than 2.1% and the guiding center radius was <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$26.6 ~\mu \text{m}$ </tex-math></inline-formula> . The design was verified by using MAGIC, EGUN, and CST simulations and good agreements were achieved.

Radial Migration from the Metallicity Gradient of Open Clusters and Outliers

Abstract Radial migration is an important process in the evolution of the Galactic disk. The metallicity gradient of open clusters and its outliers provide an effective way to examine this process. In this work, we compile metallicity, age, and kinematic parameters for 225 open clusters and carry out a quantitative analysis of radial migration via the calculated migration distances. Based on clusters with age &lt;0.5 Gyr, we obtain a present-day metallicity gradient of −0.074 ± 0.007 dex kpc−1. Three sequences are distributed along this gradient, and clusters in the upper, middle, and lower groups are found to be old outward migrators, in situ clusters, and inward migrators, respectively. The migration distance increases with age, but the time when it is most effective is probably less than 3 Gyr. The metallicity gradient breaks at guiding center radius R g ∼ 11.5 kpc, which is caused by the lack of young open clusters in the outer disk and the presence of old outward migrators in the upper sequence. This shows that this boundary is related to the different effects of radial migration between the inner and outer disks. We also found many special open clusters in and near the outer disk with R &gt; 11 kpc and a steeper metallicity gradient from the inner disk of R g &lt; 7 kpc, which tells of a complicated evolution history of the Galactic disk caused by different effects of stellar radial migration.

Experimental Generation of 1.1-kA Gyrating Electron Beam Current From an Explosive Emission Cathode Magnetron Injection Gun

Most high-power gyrotrons are driven by the thermal cathodes, and the output power can only reach megawatt (MW) level. In the present study, a single anode explosive emission cathode (EEC) magnetron injection gun (MIG) with the high beam current, the pitch radio of 1.2 and the controlled beam velocity spread was designed. At the peak voltage of 300 kV, the peak beam current of 1.1 kA and pulsewidth of 170 ns were emitted by a carbon fiber array (CFA) emitter. Compared to the reported EEC MIG, large beam loss was avoided in this experiment. The guiding center radius of 14.1 mm was close to the simulation results. CFA has the stronger emission capability, the lower emission threshold electric field, and the better emission uniformity than sharpened edge graphite (SEG), so it is suggested as the emitter of the EEC MIG.

A High-Current-Density Terahertz Electron-Optical System Based on Carbon Nanotube Cold Cathode

A high-current-density terahertz electronoptical system based on a carbon nanotube (CNT) cold cathode has been investigated in order to solve notable mode competition in high-order harmonic gyrotrons. Simulation results show that a near-axis electron beam can be generated, with a beam current of 160 mA at an accelerating voltage of 23.5 kV. The source supports the electron beam with a velocity ratio of 1.1 and a guiding center radius of 57 μm. A narrow beam spatial distribution is evidenced by adjusting control anode voltage, satisfying the need for high operating current density in slow wave devices. The current density of the electron beam is up to 620 A/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> at the output port and the velocity ratio is 0.44, with parallel energy accounting for 84% of the total energy.

Design and simulation of a W‐band gyrotron oscillator based on self‐consistent nonlinear theory

AbstractBased on the self‐consistent nonlinear theory, a W‐band continuous wave 30 kW fundamental harmonic gyrotron oscillator is designed and simulated. With a 30.0 kV, 3.0 A electron beam, the output power of more than 35 kW is obtained and the maximum electronic efficiency is equal to 40.3%. The effects of electron beam voltage, beam current, magnetic field intensity, guiding center radius, and velocity spread on electron efficiency and output power are analyzed in detail. In addition, the images of correlative electron clustering states in the process of beam‐wave interaction are presented intuitively.

Numerical Research on Mode Competition of a Coaxial Corrugated Gyrotron With Two Electron Beams

A coaxial corrugated gyrotron with two electron beams is investigated on the mode competition. Its time-dependent multimode competition equations and dependence of the starting current on the static magnetic field and the current ratio are given under the two electron beams having unequal currents. On this basis, the influence of the two electron beams having different guiding center radii and currents on the multimode competition is studied by numerical calculation. Research studies show that under the total current remaining constant, the gyrotron has different startup scenarios when two electron beams with unequal currents locate at the different guiding centers. Compared with the gyrotron with one electron beam, the beam–wave interaction efficiencies of the two electron beams show a different increase. Results show that for a 170-GHz coaxial corrugated gyrotron with two electron beams, under a total current, $I$ , of 125 A, beam voltage, $U$ , of 88 kV, static magnetic field, B, of 6.81 T, pitch factor, $\alpha $ , of 1.35: 1) when one electron beam locates at $R_{\mathrm{ e}}^{(1)}=11.15$ mm and the other at $R_{\mathrm{ e}}^{(2)}=14.32, 15.56$ mm, and $K=I_{1}/I=0.38, 0.47$ , the $\mathrm {TE}_{38,20-}$ mode can depress the startup of its competing modes; when $R_{\mathrm{ e}}^{(1)}=11.15$ mm, $R_{\mathrm{ e}}^{(2)}=12.95$ mm, $K=I_{1}/I=0.56$ , the $\mathrm {TE}_{35,21+}$ mode can also depress the startup of its competing modes and 2) when the gyrotron operates steadily in the $\mathrm {TE}_{38,20-}$ mode, it has the output power and efficiency, respectively, of 4.12 MW and 38.13% at 170.07 GHz under $K=0.38$ .

Design of second harmonic terahertz gyrotron cavity based on double confocal waveguide

Quasi-optical confocal cylindrical waveguide possesses a lot of good characteristics, such as big power capacity and low mode density, which can suppress the mode competition in beam-wave interaction. So quasi-optical waveguide has a great advantage in designing high harmonic terahertz gyrotrons. For the reason that part of electron beams located in a region of weak field intensity play a limited role in beam-wave interactions, the beam-wave interaction is not efficient in confocal cavity. Motivated by enhancing the beam-wave interaction efficiency of quasi-optical gyrotron, we propose a novel terahertz harmonic gyrotron cavity with double confocal waveguide in this paper. The transverse field distribution and the mode spectrum in double confocal waveguide are analyzed and presented. A 330 GHz second harmonic gyrotron with double confocal cavity is designed, theoretically analyzed and simulated by using a particle-in-cell (PIC) code. The results obtained for double confocal cavity are compared with the results for single confocal cavity, and the physical mechanism of beam-wave interaction enhancement in double confocal cavity is discussed. Theoretical results show that the double confocal cavity is able to increase the coupling strength of beam-wave interaction, thus, to improve the output power and the interaction efficiency of quasi-optical gyrotron. The PIC simulation results suggest that a high-order waveguide mode in double confocal cavity can steadily interact with the high harmonic cyclotron mode of electron beam without mode competition. Driven by a 40 kV, 2 A electron beam with a guiding center radius of 1.65 mm and velocity ratio equal to 1.5, output power of 9.9 kW at 328.93 GHz can be generated in the designed double confocal cavity. The beam-wave interaction efficiency increases from 5.3% in single confocal cavity to 12.4% in dual confocal cavity under the same operation parameters. The double confocal cavity has great potential applications in terahertz band. Moreover, this study indicates that the eigen mode in double confocal waveguide is a kind of hybrid mode superimposed by two independent single confocal waveguide modes. This mode characteristic will be beneficial to designing a multifrequency gyrotron oscillator operated in two modes and two cyclotron harmonics, simultaneously, with a single electron beam used, which provides a new possibility to develop the novel terahertz radiation source.

Triode Type Coaxial Inverse Magnetron Injection Gun for 2-MW, 240-GHz Gyrotron

A triode type coaxial inverse magnetron injection gun (CIMIG) is designed using the 2-D and 3-D beam trajectory simulation codes for 240-GHz, 2-MW coaxial gyrotron. In IMIG, the modulating anode is adjusted at the center of the emitter ring and it provides compact size and better control on the beam instability and beam halo. As per the beam-wave interaction simulations, the CIMIG is designed to deliver the 6.5-MW (100 kV, 65 A) helical electron beam of 8.72-mm guiding center radius at the cavity center. First, the synthesis of CIMIG is performed to estimate a range of electrical and geometrical parameters based on the tradeoff equations. The Gaussian type of magnetic field profile is optimized with the peak magnetic field value of 10 T at cavity center and the beam compression ratio of 35.11. The pitch factor of 1.306 and the transverse velocity spread of 2.74% are realized in 2-D trajectory code electron GUN. The design results obtained in 2-D simulations are validated in 3-D particle trajectory simulator CST-Particle Studio. In 3-D optimization, the gun geometry and the electrical parameters are kept similar to 2-D design. The pitch factor of 1.285 and transverse velocity spread of 3.01% are obtained in 3-D design which shows good agreement with the 2-D design.

Inverse Magnetron Injection Gun for Megawatt Power, Sub-THz Gyrotron

Inverse magnetron injection gun (IMIG) provides several advantages over simple MIG especially in high-power, high-frequency gyrotron. Sub-millimeter (>230 GHz), high-power (≥1 MW) gyrotrons would be required in futuristic plasma fusion machines to fulfill the need of tens of megawatt RF power for electron cyclotron resonance heating. Here, in this paper, an IMIG is designed for 0.24-THz, 1-MW gyrotron. First, the tradeoff design equations and technical constraints are used to synthesize the electrodes geometry and electrical parameters of IMIG. This synthesized IMIG model is further simulated to finalize the design using electron trajectory code EGUN. Simultaneously, the magnet system including the auxiliary gun coils and superconducting magnets (SCM) is also designed and the obtained magnetic field profile is used in the IMIG simulations. In the case of high-power MIG, the suppression of the beam halo is a major design concern and analyzed critically in the present IMIG design. The shape of the modulating anode and main anode is optimized to make a beam halo shielding. This IMIG is designed for a very high-order mode (TE46,17) simple cylindrical cavity, which confines the range of the tolerance in the electron beam parameters. So, a detailed parametric and misalignment analysis is also performed with respect to the electron beam parameters, such as pitch factor, velocity spread, and guiding center radius spread. These parametric and misalignment analyses would be utter helpful in the fabrication, assembling and testing of the IMIG as well as gyrotron.

On the Kinematic Signature of the Galactic Warp As Revealed By the LAMOST-TGAS Data

Abstract Using a sample of about 123,000 stars with accurate 3D velocity measurements from the LAMOST-TGAS data, we confirm the kinematic signature of the Galactic warp recently found by Schönrich &amp; Dehnen. The data reveal a clear trend of increasing mean vertical velocity as a function of absolute vertical angular momentum L z and azimuthal velocity V ϕ for guiding center radius R g between 6.0 and 10.5 kpc. The trend is consistent with a large-scale Galactic warp. Similar to Schönrich &amp; Dehnen, we also find a wave-like pattern of versus L z with an amplitude of ∼0.9 km s−1 on a scale of ∼2.0 kpc, which could arise from bending waves or a winding warp. Finally, we confirm a prominent, localized peak in near L z ∼ 2150 kpc km s−1 (corresponding to R g ∼ 9 kpc and V ϕ ∼ 255 km s−1). The additional line-of-sight velocity information from LAMOST reveals that stars in this feature have a large, inward radial velocity of V R ∼ −13.33 ± 0.59 km s−1 and a small radial velocity dispersion of σ R ∼ 25.27 ± 0.89 km s−1, suggesting that a stellar stream gives rise to this feature.

Study on the Effect of Electron Beam Quality on a Continuously Frequency-Tunable 250-GHz Gyrotron

The effect of the electron beam quality, including the variation of the velocity spread, the guiding center radius spread, and the pitch factor when the operating frequency of a 250-GHz TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">52</sub> mode continuously frequency-tunable gyrotron used for dynamic nuclear polarization (DNP) enhanced nuclear magnetic resonance (NMR) spectroscopy is tuned by changing the operating voltage V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> or the operating magnetic field B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , on the operating frequency, the frequency-tunable range and the beam-wave interaction efficiency, has been studied based on a selfconsistent nonlinear beam-wave interaction theory. It is found that the velocity spread changes from 2.2% to 7.3% and the guiding center radius spread varies from 0.6% to 1.5%, when the operating voltage V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> changes between 8 and 15 kV at an operating magnetic field of 9.24 T, the velocity spread varies from 4.1% to 6% and the guiding center radius spread changes from 0.8% to 1.4%, when the operating magnetic field B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> changes between 9 and 9.3 T at an operating voltage of 10.1 kV. The variation of the velocity spread and guiding center radius spread with respect to the operating voltage or the magnetic field have a significant effect on the beam-wave interaction efficiency and the frequency-tunable range of the 250-GHz TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">52</sub> mode continuously frequency-tunable DNP/NMR gyrotron,and the effect of the variation of the velocity spread and the guiding center radius spread should be taken into account in a frequency-tunable gyrotron design.

Design of a Double Anode Magnetron Injection Gun for Q-band Gyro-TWT Using Boundary Element Method

This paper presents a novel design code for double anode magnetron injection guns (MIGs) in gyro-devices based on boundary element method (BEM). The physical and mathematical models were constructed, and then the code using BEM for MIG’s calculation was developed. Using the code, a double anode MIG for a Q-band gyrotron traveling-wave tube (gyro-TWT) amplifier operating in the circular TE01 mode at the fundamental cyclotron harmonic was designed. In order to verify the reliability of this code, velocity spread and guiding center radius of the MIG simulated by the BEM code were compared with these from the commonly used EGUN code, showing a reasonable agreement. Then, a Q-band gyro-TWT was fabricated and tested. The testing results show that the device has achieved an average power of 5kW and peak power ≥ 150 kW at a 3% duty cycle within bandwidth of 2 GHz, and maximum output peak power of 220 kW, with a corresponding saturated gain of 50.9 dB and efficiency of 39.8%. This paper demonstrates that the BEM code can be used as an effective approach for analysis of electron optics system in gyro-devices.

Theoretical investigation on a multifrequency multimode gyrotron at &lt;italic&gt;Ka&lt;/italic&gt;-band

In this paper, a multifrequency, multimode gyrotron has been designed, which can operate at 28, 29, 31, and 32 GHz with corresponding modes TE0.3, TE+5.2, TE−3.3, and TE+6.2, respectively. For all operating status, the mode competitions have been investigated carefully with the help of a new time-dependent, multimode, self-consistent code, which is built on the trajectory approach. For the operating at 28 GHz, it also has been simulated by CST Particle Studio. They have similar results when comparing two results. For the operating at other status, the changes of electron parameters caused by the changed dc-magnetic field have been analyzed in detail. In the analyzes, the guiding center radius and Larmor radius would not be influenced severely, but the velocity ratio would be influenced seriously, such as, the velocity ratio of operating at 32 GHz would be reduced to about 0.9787, which caused a severe reduction in the efficiency. In order to alleviate the influences, a compensation magnetic coil (CMC) has been used in the magnetron injection gun region, in which the changes of electron parameters have been analyzed too. The simulation results show that the electron efficiency of operating at 32 GHz can be increased from 24.6% to 31% by applying CMC. The multimode multifrequency gyrotron operated at ka -band gyrotron has been investigated in theory, which can provide new possibilities in high-power millimeter source development.

A gyrating electron beam source for frequency tunable, 200 GHz gyrotron

Magnetron injection gun (MIG) is an important and critical part of any gyrotron. The overall performance of gyrotron deeply depends on the quality of gyrating electron beam. In this article, the design of triode type magnetron injection gun for 200 GHz frequency tunable gyrotron is discussed. It is planned to design and develop a 200 GHz gyrotron for 300 MHz DNP-NMR spectroscopy. The triode MIG is designed in such a way that it can be used in 200 GHz gyrotron of higher RF power for other applications also such as electron spin resonance, imaging. First, geometrical and electrical parameters of MIG are calculated using trade-off and scaling equations. Based on the calculated parameters, further, electron beam trajectory simulations are performed in EGUN. The geometry of electrodes are optimized through numerous simulation iterations to achieve the required beam parameters, such as, minimum spread in velocity and guiding center radius, an optimum value of velocity ratio (1.3–1.5), guiding center radius as per the maximum beam-wave coupling strength. The magnet system is designed to achieve the smooth Gaussian type of magnetic field profile with peak value of 7.35 T at cavity center. Further detail parametric analyses are performed which would be helpful in the frequency tuning and long-time stable operation of gyrotron device.

ON THE CHEMICAL AND STRUCTURAL EVOLUTION OF THE GALACTIC DISK

We study the detailed properties of the radial metallicity gradient in the stellar disk of our Galaxy to constrain its chemical and structural evolution. For this purpose we select and analyze $\sim$ 18,500 disk stars taken from two datasets, the Sloan Digital Sky Survey (SDSS) and the High-Accuracy Radial velocity Planetary Searcher (HARPS). On these surveys we examine the metallicity gradient, $\Delta$[Fe/H]/$\Delta R_{\rm g}$, along the guiding-center radii, $R_{\rm g}$, of stars and its dependence on the [$\alpha$/Fe] ratios, to infer the original metallicity distribution of the gas disk from which those stars formed and its time evolution. In both sample sources, the thick-disk candidate stars characterized by high [$\alpha$/Fe] ratios ([$\alpha$/Fe] $>$ 0.3 in SDSS, [$\alpha$/Fe] $>$ 0.2 in HARPS) are found to show a positive $\Delta$[Fe/H]/$\Delta R_{\rm g}$, whereas the thin-disk candidate stars characterized by lower [$\alpha$/Fe] ratios show a negative one. Furthermore, we find that the relatively young thin-disk population characterized by much lower [$\alpha$/Fe] ratios ([$\alpha$/Fe] $<$ 0.2 in SDSS, [$\alpha$/Fe] $<$ 0.1 in HARPS) shows notably a flattening $\Delta$[Fe/H]/$\Delta R_{\rm g}$ with decreasing [$\alpha$/Fe], in contrast to the old one with higher [$\alpha$/Fe] ratios ([$\alpha$/Fe] $\sim$ 0.2 in SDSS, [$\alpha$/Fe] $\sim$ 0.1 in HARPS). The possible implication for early disk evolution is discussed, in the context of galaxy formation accompanying the rapid infall of primordial gas on the inner disk region, which can generate a positive metallicity gradient, and the subsequent chemical evolution of the disk, which results in a flattening effect of a metallicity gradient at later epochs.

Multimode steady-state theory for Gyro-TWT and simulation of mode competition

In this paper, a multimode steady-state theory for Gyro-TWT is established. Effects of the input power and guiding center radius on backward wave oscillation in a TE11 Gyro-TWT are studies by this theory. The result shows that either of the two effects can reduce the output power of the backward wave oscillation to a certain extent, while both of them simultaneously can more effectively improve the output characteristics and the stability of the TE11 mode.

AN ANALYTIC THEORY FOR THE ORBITS OF CIRCUMBINARY PLANETS

Three transiting circumbinary planets (Kepler-16 b, Kepler-34 b, and Kepler-35 b) have recently been discovered from photometric data taken by the Kepler spacecraft. Their orbits are significantly non-Keplerian because of the large secondary-to-primary mass ratio and orbital eccentricity of the binaries, as well as the proximity of the planets to the binaries. We present an analytic theory, with the planet treated as a test particle, which shows that the planetary motion can be represented by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the binary's potential, the epicyclic motion, and the vertical motion. In this analytic theory, the periapse and ascending node of the planet precess at nearly equal rates in opposite directions. The largest forced oscillation term corresponds to a forced eccentricity (which is an explicit function of the parameters of the binary and of the guiding center radius of the planet), and the amplitude of the epicyclic motion (which is a free parameter of the theory) is the free eccentricity. Comparisons with direct numerical orbit integrations show that this analytic theory gives an accurate description of the planetary motion for all three Kepler systems. We find that all three Kepler circumbinary planets have nonzero free eccentricities.