Musk says the orbital heat shield is currently SpaceX's biggest problem with making the Ship fully reusable, which is necessary to make Starship a fully reusable rocket.
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Upon reentry into Earth's atmosphere, the rocket travels at extreme speeds through Earth's dense atmosphere. This resistance with the atmosphere causes friction, and just like a meteor, the increase in temperature eventually causes flames. This is where the orbital heat shield comes in.
SpaceX has equipped its rockets with a shield designed to disperse the heat generated by reentry, but unfortunately, no one has made a fully reusable heat shield. Musk briefly discussed what the orbital heat shield is made of, saying it needs to be constructed of extremely fine strands of glass and aluminum oxide (sapphire) fibers organized in a very specific geometry for the highest efficiency.

Heat shield tiles being tested
In addition, "special coatings" are added to increase the heat shield's resistance to the generated heat and give it structural integrity, as the design is quite brittle. Musk said the heat shield's brittleness is similar to a coffee cup, and SpaceX requires it not to crack or break, as that would mean the chance of the Ship being damaged, the structural integrity of the rocket being compromised, and ultimately, by definition, rocket not being reusable. This problem has proven extremely difficult since the rocket shakes when launched/reentering Earth's atmosphere, which can cause the orbital heat shield tiles to break.

Musk was asked if development is being done on the materials technology for the heat shield, and he said, "Yes, it's a very difficult problem. A problem that no one has ever solved." Musk says SpaceX needs to get the exact materials combination correct, with the right molecules aligned in the right shape, to achieve a fully reusable heatshield, which then must be applied to the rocket with zero mistakes.

Another problem SpaceX is facing with the orbital heat shield tiles with their attachment on the rocket is the expansion and contraction of the structure unnearth. Musk says the expansion/contraction is different depending on specific regions of the rocket and used the cryogenic fuel tank as an example. According to the SpaceX CEO, when fuel is loaded into the tanks, they will contract because it is extremely cold. However, upon re-entry, when the rocket is increasing in temperature, the tanks will expand.
This expansion and contraction of the structure underneath the tiles also expands and contracts the gap between the orbital heat shield tiles, with a difference of approximately 10-20% between post and pre-launch. Because of this contraction/expansion problem, the orbital heat shield tiles can't be placed end to end as they would simply crack.