Yes, a Radeon HD 7700 will run Horizons. However, if you are borderline on performance now, you will most likely need to reduce graphics settings to get acceptable frame rates on planet surfaces.
EXPECTED PERFORMANCE
Our minimum requirements aim to produce a minimum of 30 frames-per-second at 720p (1280x720).
Our recommended requirements aim to produce an average of 60 frames-per-second at 1080p (1920x1080). This may fluctuate in more intensive areas and on planets (Horizons-only)