As far as I know, no institute in India offers astronomical engineering. Tata Institute of Fundamental Research also doesnt conduct any such course. However Astronomy and Astrophysics programmes are at the following

Indian Institute of Astrophysics, Bangalore (IIA),

IndianInstitute of Science, Bangalore (IISc),

National Centre for Radio Astrophysics, Pune (NCRA-TIFR),

Inter-University Centre for Astronomy and Astrophysics, Pune (IUCAA),

Physical Research Laboratory, Ahamedabad (PRL),

Raman Research Institute, Bangalore (RRI) and

Tata Institute of Fundamental Research, Mumbai (TIFR

These are mostly at PhD level and research oreinted. However IISc Bangalore offers an integrated course called JAP - Joint Astronomy programme in association with the other institutes mentioned above. All details for JAP is mentioned here. http://www.physics.iisc.ernet.in/~jap/jap-home.htm.

I dont know if you have completed your graduation or not. If not, you should first complete that before you think of specializing in astronomy.

There are following branches in TIFR under Physics (School of Natural Sciences)

Astronomy and Astrophyics

High Energy Physics

Theoritical Physics

Nuclear and Atomic Physics

Rest you can check the details below:

Courses

Good Luck.

The cutoff is not decleared on its official site for B.Sc Physics.

Hi, You have not mentioned the course you want to take admission in. I am sharing an official link which states eligibility criteria of various courses.

Check here: TIFR Eligibility

All the best!