The tenth House is associated with career, profession and social status - position in society. The Sun represents our drive to express ourselves through creating something, preferably something that lasts - its our creativity and our will power.
A reasonable supposition would therefore be that someone with the Sun in the tenth has a desire to express their creativity through their career or through achieving social status. This also suggests that such a person will be ambitious and desire success in their chosen profession or career - without it their Sun cannot express itself.
Whether the desire to succeed is accompanied by an ability to handle their ambition wisely and to direct it well, is shown by other areas of the chart but would be one issue that would be considered in a reading. You might look at the Martha Stewart thread, as she has her Sun conjunct her MC.
The 'Public' are signified by the Moon, so a public scandal is likely to involve the Moon in some way - by sign, aspect and house position or through transit or progression. If this scandal is connected to their ambition and drive for success then it is likely that the Sun and Moon will be linked by Aspect in the chart or by transit or progression.
