CORP SMT DEPARTS PRIOR TO ALLOWING TIME FOR GYROSCOPES TO SPOOL UP; CAUSING HDG DEV.

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I FILED AN IFR FLT PLAN TO DEPART AN UNCTLED ARPT 13.5 KM SW OF HOUSTON HOBBY BY PHONE. I LOADED PAX AND CONTACTED ATC IN THE AIR. DUE TO THE RAPID ENG START; TAXI; AND TKOF; THE SLAVED GYROSCOPE FOR THE HDG INDICATOR WAS GROSSLY OFF. THE INST VACUUM SYS (KNOWN TO BE WEAK BY PLT) WAS SLOW TO SPOOL UP GYROSCOPES. WHEN RADAR IDENTED; ATC ASSUMED I WAS TURNING IN THE WRONG DIRECTION TO THE CORRECT HDG. I BEGAN TO QUERY IN MY MIND THE INITIAL ATC HDG AND ALT BECAUSE OF A 2000 FT ANTENNA FARM (LOCATION KNOWN TO PLT) NEAR MY FLT PATH. I CONCENTRATED ON A VX CLB WHEN A HIGHER ALT WAS INITIALLY GIVEN ON RADAR CONTACT. I TOLD ATC; 'I HAVE THE TWRS;' AND WAS GIVEN A HDG CHANGE OF 90 DEGS. 10 SECONDS LATER; ATC ASSUMED AND WAS CORRECT IN STATING MY DIRECTIONAL GYROSCOPE WAS GROSSLY OFF. AN IMMEDIATE; 'R TURN;' WAS STATED BY ATC. ATC WAS ENGROSSED WITH ME AND WAS VECTORING NUMEROUS ACFT FOR THE HOBBY ARPT. ONCE ATC ESTABLISHED MY DIRECTIONAL GYROSCOPE WAS PROPERLY SET AND OPERATING; VECTORS AND ALTS WERE GIVEN TO RESOLVE CONFLICTS. PLT HAS FLOWN OUT OF DEPARTING ARPT OVER 200 TIMES IN THE LAST YR AND TRIED TO COMPLY WITH A PAX REQUEST TO 'HURRY.' CORRECTIVE ACTION FOR FUTURE: THE PIC IS RESPONSIBLE FOR ALL ASPECTS OF THE FLT AND MUST NOT ALLOW HIMSELF TO BE INTIMIDATED BY PAX OR ATC INTO DOING SOMETHING IN AN UNSAFE MANNER IN THE MIND OF THE PLT.