Making plane engines is primarily aircraft propulsion and aviation, which come under the categories of aeronautical engineering (aeronautics). Aeronautical engineering shares quite a lot of things in common with astronautical engineering, and so together they form aerospace engineering.
As far as physics goes, you need pretty much the whole syllabus! Certainly thermodynamics, classical mechanics, aerodynamics, rockets and flight.
Thermodynamics (heat, energy transfer), condensed matter (crack propagation, material study), mechanics (friction of the system, propulsion/thrust, weight).
If you can I would suggest some form of design software training as well, once you’ve got the theory down designing them will require specialist software.
If you intend a career in aeronautical engineering, then the answer to this is that you should study aeronautical engineering or aerospace engineering, not physics.
If you want to know what the physics underlying the design of aeroplane engines is, then the answer is, quite a lot! The branch of physics known as thermodynamics basically arose out of steam engine development in the 19th century, and is still crucial to engine design. The physics of materials is important in understanding what to make your plane engine out of (which materials will withstand the heat and stresses endured by the blades of a jet turbine, for example?), and how to design it (the early British Comet jet airliner suffered several catastrophic accidents in which the plane broke up in midair, which were eventually traced to the fact that it had square windows instead of oval ones: the extra stress at the corners generated cracks, which then propagated outward and eventually caused the whole fuselage to break apart). Aerodynamics determines what shape your engine should be to minimise drag, and also, applied to the whole plane, how much power you need to provide. These days, acoustics is also an important subject: you want to minimise the noise nuisance from your plane. Fluid mechanics is essential for understanding the airflow round and through your engine. I will admit that particle physics, relativity and astrophysics are unlikely to have much impact on the design of aeroplane engines, but other than that pretty much the entire degree content is relevant.
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