What is the speed that the earth and solar systems is moving through the unviverse?

The plane of the Solar System is tilted about 90° with the plane of the Galaxy. The Solar System is in the Orion Arm (called also Orion Spur), inner with respect to Perseus Arm. This motion has a speed of 217.215 Km/s, hence the Solar System takes 226 million years to complete one lap around the Galactic Nucleus.

The movement onward-backward the nucleus of the Milky Way is determined by the gravitational pull of the nucleus and the stars situated inner the orbital track of the Solar System and by the gravitational pull of the stars outer from the track of the Solar System. This movement has a speed of 20 Km/s. The motion upwards-downwards the plane of the galaxy are influenced by the motions of the bodies that constitute the Solar System, including the motions of the Sun. The speed of this movement is 5-7 Km/s and comprehends about 20 light years.

The Cosmic Cloud is situated up and ahead the Solar System. The Cosmic Cloud is going onwards-backwards the nucleus of the Milky Way at 15-20 Km/s. It is approaching the Solar System at a relative speed of 37 Km/s. We think our Solar System will encounter the cloud at any moment in the next years. Right now, we are experiencing some large “puffs” of dust and cosmic radiation, but those are not part of the main cloud.

The motion of the cosmic clouds and the interstellar material that form the arms fo the galaxy are quite apart from the movements of the stars that are originated in those clouds.

There are other large "bubbles" of Cosmic Radiation (Interstellar Wind) that will shock with the Solar Wind, causing changes in the activity of the Sun and in the climate of the Planets of the Solar System. The starships Voyager-1 and Voyager-2 have detected high densities of Cosmic Radiation that are affecting the climate on Earth and other planets of the Solar System. The changes have been manifested like increases of the Tropospheric Temperature on Earth above the standard fluctuations. The warming and the climate change have been detected also in Venus, Mars, Jupiter, Saturn, Uranus and Pluto. Some satellites of Jupiter and Saturn are experiencing global warming and climate change.

There is no such thing as speed without specifying a frame of reference. What are you intending to measure this speed against? It cannot be the center of the universe, or point of the big bang, because there is no such thing. The big bang happened, is happening, everywhere.

About 600 km/s with respect to the reference frame of the cosmic microwave background.

360 degrees in 24 hours is i dont know in speed

soo larage