Short answers, pulled from the story.
Caliph al-Ma'mun commissioned the first major observatory in Baghdad in the early 9th century. This facility became the engine of a scientific revolution where astronomers measured the circumference of the Earth with unprecedented precision. The House of Wisdom housed this research and synthesized Greek, Persian, and Indian knowledge into a new Islamic science.
Ibn Yunus discovered critical errors in Ptolemy's calculations regarding the Earth's axial precession by calculating the rate of change to be one degree every 70 years. This finding contradicted the previously accepted rate of 100 years. His work contributed to a quiet rebellion within Islamic astronomy during the 10th century.
The astrolabe served as a portable model of space that allowed users to calculate the approximate location of any heavenly body found within the Solar System at any point in time. It enabled users to determine the qibla, tell time, and navigate the seas by accounting for the latitude of the observer. A later iteration known as the Mariner's astrolabe was developed to counteract difficult sea conditions.
Nasir al-Din al-Tusi supervised the technical construction of the observatory at Maragha in 1259 under the patronage of Hulagu Khan. This facility became the crucible for a new astronomical tradition that included resting quarters for the Khan, a library, and a mosque. The collaboration resulted in important modifications to the Ptolemaic system over a period of 50 years.
The Persian astronomer Jamal ad-Din presented seven Persian astronomical instruments to Kublai Khan in 1267. These instruments included a terrestrial globe, an armillary sphere, and an astronomical almanac known in China as the Wannian Li. Jamal ad-Din was appointed in 1271 as the first director of the Islamic observatory in Beijing.
The observatory founded by Taqi ad-Din Muhammad ibn Ma'ruf in Constantinople was destroyed in 1580. Although founded in 1577, the facility was short-lived because opponents of prognostication from the heavens successfully sought its destruction. This event marked a turning point in the history of Islamic astronomy and demonstrated the vulnerability of scientific institutions to political and religious pressures.