Been studying silk road history. Came across this interesting topic. Throughout the history of nomadic empires we often see legitimization efforts proceed conquest. Thus we get the patronage of things like religion, monumental architecture, and biographical literature.
But interestingly enough under Hulagu Khan and Kubulai Khan, we get the patronage of astronomy in Persia and China respectively. We see the construction of massive observatories like at Maragheh, and the movement of famous astronomers like al-Tusi and Jamal ad-Din. Leading up to the famous Timurid prince Ulugh Beg creating the Samarkand observatory.
Why would these nomadic powers patronize astronomy? What would they get out of it?
The Mongols and subsequent dynasties which claimed to be relatives of the Mongols (Tamerlane) patronized the exact sciences. As you have already mentioned Ulugh Beg was the grandson of Temür or Tamerlane as the western world calls him. He built the grand observatory in Samarkand. In addition, Nasir al-Din Tusi was one of Hulegu's advisers. Hulegu was the leader of the Ilkhanate in Persia. Nasir al-Din Tusi was also the lead astronomer at the Maragheh Observatory. The above referenced groups recieved several things from the study of astrology:
First, prediction of eclipses. The traveler Juvaini recorded that the shamans of the Mongolian tribes could predict when an eclipse of the sun or moon would occur.
When one is about to take place all the people lay in their food, for they must not go out of the door of their dwelling. And while the eclipse is taking place, they sound drums and instruments, and make a great noise and clamor. After the eclipse is over, they give themselves to drinking and feasting, and make great jollity. It is hard to tell from this passage what exactly the Mongols thought about eclipses, but you can guess that they probably thought that eclipses were unlucky days. It seems like the Mongols would stay indoors for fear of something bad happening. (Spuler, 110)
Second, the prediction of luck and unlucky days. Again, Juvaini writes a passage about the shamans predicting which days were good to make war and which days were not good to make war. Juvaini wrote that the Mongols would not begin a war or end a war without the consent of the shamans. (Spuler, 111)
Third, the new and more accurate calendar. Nasir al-Din Tusi was one of the leading Muslim astronomers of his time. He created star charts and wrote an astronomical manuscript. He used mathematics to calculate the precession of the equinoxes. This allowed the Mongols of Kublai Khan's Yuan dynasty to create a more accurate calendar. You can reason that this is probably good for trade and the economy of the society. (Kwanten, 224) (Ballay, 391)
Fourth, the prediction of accurate weather patterns and better geographic maps. Rossabi claims that the Mongols often tied astronomy and astrology together. This resulted in advances in both weather prediction and geographic maps.
Astronomy offered benefits in predicting climactic and weather patterns, information which could prove vital to the economy...Greater geographic knowledge and more precise maps contributed to safer travel and an expansion of trade throughout Asia. In essence, both of these things impacted trade and the economy of the Yuan dynasty in particular. (Rossabi, xvi)
So, in the end, the Mongols and subsequent Mongol dynasties received several benefits from the study of astronomy. Most of which impacted trade, conflict, and spiritual beliefs.
Sources: Luc Kwanten's Imperial Nomads, 1979.
Bertold Spuler's History of the Mongols, 1968.
"The Astronomical Manuscripts of Naṣīr al-Dīn Ṭūsī" by Ute Ballay in Arabcia, 1990.
Morris Rossabi's Eurasian Influences on Yuan China, 2013.