Long-duration gamma-ray bursters are only observed in galaxies where there is ongoing star formation. Explain how this is consistent with the collapsar model of how these bursters occur.

The collapsar model is a cornerstone in our understanding of the most powerful explosions in the universe. Imagine the life of a massive star as a battle against gravity, with nuclear reactions at its core providing the firepower. When the star's nuclear fuel is depleted, it can no longer fight against its own gravitational pull. The core collapses and, if the star is massive enough, a black hole is born in an explosive event called a 'collapsar'.

During this collapse, the spinning star's outer layers fall toward the newly-formed black hole, creating a rapidly rotating accretion disk. This infalling material and the immense gravitational energy convert into an astoundingly brilliant show of gamma rays, known as a long-duration gamma-ray burst. It's like watching nature’s grand finale of a star's life, where the star gives one last magnificent performance that can be detected across the cosmos.

Star formation is the cosmic process that serves as the universe's nursery. Clouds of gas and dust, primarily hydrogen, coalesce under gravity to create the cradles of infant stars known as stellar nurseries. As the material in these clouds condenses, it heats up and ultimately triggers nuclear fusion, giving birth to a new star. This is an ongoing cycle, rather like a garden that continually sprouts new flowers.

These regions are spotted throughout galaxies, particularly within their spiral arms or dense molecular clouds. Turbulence within the clouds leads to knots of denser material where stars can form. Observing various galaxies with active star-forming regions gives us a peek at stellar nurseries, helping us understand that stars come into existence in all stages simultaneously.

The life of a star is an epic tale of transformation. Stars, like people, are born, live their lives, and eventually die, but their lifespans span millions to billions of years. The initial mass of the star dictates its life cycle. Massive stars lead short, dramatic lives, while smaller stars, such as our Sun, enjoy a steadier pace and a longer lifespan.

Stars progress through stages that include main sequence, red giant or supergiant phase, and eventually a final death throes phase which could be a supernova explosion for massive stars. At the end of their life, their cores collapse, leading to either a neutron star or black hole, depending on the original mass of the star. Each stage of this cycle offers unique clues to astronomers about the workings of the universe.

Black holes are among the most mysterious and fascinating objects in space. They are regions where gravity is so strong, nothing, not even light, can escape their grasp. This intense gravity occurs because a massive amount of matter has been squashed into a tiny area at the end of a star’s life. The result is a gravitational pull that warps space and time.

Black holes are not visible using traditional telescopes, but they can be detected by watching for their effects on nearby stars and gas. When a star comes too close, it can be torn apart and sucked in, which is called 'spaghettification'. This process heats up the star's material and causes it to emit X-rays, which we can detect and use to infer the presence of a black hole.