Hercules Stream and the Outer Elliptical Ring R₁ₑ_₂

We study the formation of the Hercules stream in the model Galactic disk which includes the outer resonance ring R₁R₂ located near the Outer Lindblad Resonance (OLR) of the bar. The Hercules region and the anti-Hercules region, introduced for calibration, were restricted in space by the solar neighbourhood r<0. 5 kpc, and on the (Vₑ, Vₓ) plane by ellipses centered at Vₑ=25 km s ^-1 and Vₓ=200 km s ^-1 (Hercules), and at Vₑ=-25 km s ^-1 and Vₓ=200 km s ^-1 (anti-Hercules). The number of stars in the Hercules region reaches a maximum in the time period of 2-3 Gyr from the start of simulation and the number of stars in the anti-Hercules region oscillates with a period of 1. 8 0. 1 Gyr. The majority of stars in the model disk located in the Hercules and anti-Hercules regions have orbits elongated perpendicular and parallel to the bar, respectively. The median value of the initial distances of stars in the Hercules (anti-Hercules) region is slightly smaller (larger) than the OLR radius, respectively. There are two types of orbits in the Hercules region. Orbits of the first type always lie inside a figure bounded by two ellipses elongated perpendicular to the bar. Orbits of the second type are elongated at the angles of -60^ or 60^ to the major axis of the bar most of the time. The distribution of stars in the Hercules region along the period of slow oscillations in the angular momentum has two maxima: P=0. 7 and 2. 6 Gyr corresponding to orbits of the first and second type. In the anti-Hercules region, most orbits are captured by libration relative to the major axis of the bar with a period of 1. 9 Gyr. In general, orbits in the Hercules and anti-Hercules regions support the outer rings R₁ and R₂ elongated perpendicular and parallel to the bar, respectively. Stars from the Gaia DR3 catalog located in the Hercules region appear to be, on average, brighter, bluer, and more luminous than stars in the anti-Hercules region which is probably caused by selection effects due to different distributions of these stellar samples over the Galactic latitude b.