tokens: where m p = (4/3)ρ s 3 is the particles mass orbiting with velocity A is the particle’s cross section, and is the unit vector pointing to the direction of the radiation. The particles are assumed to be spherical with uniform density equal to 1 g cm − 3 . For transparency we give the explicit form of ω H is Haumea’s rotational frequency, n H is the mean motion about the Sun, and i is the obliquity of the dwarf planet to its orbital plane. The gravitational effect of the Sun is not included in this model because it is fifth order of magnitude weaker than that of Haumea at the distance of the ring. However, Haumea’s orbit is assumed to be circular with a period of 284 yr which is included in the time dependence of the RP. Since the obliquity of the dwarf planet is about 14 deg this implies a long-term modulation in z -component of the RP. In other words, the initial position of the Sun in our calculations is the negative x direction and Haumea’s obliquity stays in this direction during the whole integration. The effect of planetary shadowing (which blocks the stellar radiation impinging on the ring) is neglected since the ring is currently not shadowed because it is in the equatorial plane of the planet which is tilted with 13.8° ± 0.5° to the orbital plane (Ortiz et al. 2017 ). Furthermore, even for longer integration, only a small fraction of the ring will be hided during Haumea’s journey around the sun. We use a 4th order Runge–Kutta numerical scheme with a fixed step size that keeps the relative energy error below 10 −8 . Equation ( 5 ) is integrated for different times in order to cover different dynamical regimes. That is, the simulations are performed for T = 10 3 , 2.5 × 10 4 , and 3.5 × 10 5 rotation periods, yielding ca. 1, 10, 1000 yr, respectively. The calculations also have been done for grain sizes, 1, 1.75, 2.5, and 5 m. The dynamical analysis involves two different setup of initial conditions. In the one hand, when studying the fine resonant structure, particles are placed equidistantly along a line ( x -axis) in radial distance 1 r 11. On the other hand, an initial ring is defined, θ ∈ [0; 2] and r ∈ [0; 5], in order to follow the particle dynamics in general. In both cases grains are initially on circular Keplerian orbits in the equatorial plane of Haumea. 3 RING CHARACTERISTICS Our first analysis explores the lifetime of the particles around the dwarf planet. The grain dynamics yields three different end-states for the particles: (i) Escape : When the semimajor axis of a particle becomes larger than 25 r s (≈ 25575 km) it is taken to be ejected. Ejections occur due the close encounters with Haumea when the particles are perturbed to hyperbolic orbits. (ii) Collision : In our model a collision occurs when the particle approaches the dwarf planet closer than its volume-equivalent radius, 816 km (0.74 r s ) (Ortiz et al. 2017 ). (iii) Bounded motion : When a particle, until the end of the integration, neither collides nor escapes
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