It is mostly for cars with turbos. The higher octane fuel burns more cleanly and there are less leftover bits of fuel in the exhaust. This is important because the manifold exhaust is what spins the turbo, which in turn pushes more air in to the engine making more power.
Anyways, turbos get really hot, and if there is unburned fuel vapor still in the exhaust, it can ignite and blow up the turbo. This is the main reason that I’m aware of.
Octane number indicates how much can you compress the air and fuel mixture without causing it ignite spontaneously. Higher octane number means you can apply higher compression. Increased compression ratio means that the same amount of air + fuel mixture will generate more force for every revolution of the engine. So to generate more power luxury cars use high compression ratio engines that requires higher octane number fuel to resist the spontaneous ignition before actual ignition by the spark plug. Some cars also use direct gasoline injection to avoid spontaneous ignition.
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