Question: A direct ancestor of the wife of Prince William was the first member of the Fairfax family to own which castle?

Context paragraphs:
Virginia State Route 123: State Route 123 (SR 123) is a primary state highway in the U.S. state of Virginia.  The state highway runs 29.27 mi from U.S. Route 1 (US 1) in Woodbridge north to the Chain Bridge across the Potomac River into Washington from Arlington.  It goes by three local names.  From its southern terminus to the Occoquan River Bridge, it is known as Gordon Boulevard.  From the Occoquan River Bridge to the city of Fairfax it is known as Ox Road.  North from Fairfax, it is known as Chain Bridge Road.  SR 123 is a partial circumferential highway in Northern Virginia that connects Woodbridge in eastern Prince William County with the independent city of Fairfax and the Fairfax County communities of Vienna, Tysons Corner, and McLean, the last the home of the National Counterterrorism Center and the Central Intelligence Agency.  The state highway also connects all of the major highways that radiate from Washington, including Interstate 95, I-66, US 29, US 50, SR 267, and the George Washington Memorial Parkway.  Furthermore, SR 123 crosses another pair of circumferential highways, I-495 and the Fairfax County Parkway, and SR 7, a major northwest–southeast highway through Northern Virginia.  The state highway is a part of the National Highway System for its entire length.

Janet Doub Erickson: Janet Ann Doub spent her early years in Boonsboro, Maryland, on farmland her father’s ancestors had settled in the eighteenth century and subsequently farmed continuously.  She moved to New England in the nineteen thirties to be closer to her mother’s family, who were descended from New England's earliest pioneers.  One direct ancestor, the aristocratic Henry Sherburne of Portsmouth, New Hampshire numbered among New Hampshire's first English settlers, arriving there in 1632, a little more than a decade after the Mayflower landed on Cape Cod.  Sherburne and his descendants settled New England and beyond as the frontier moved west.  Another maternal ancestor was Thomas Wiggin, the first governor of the province of New Hampshire.  Mid-Atlantic, her father's Maryland forebears were the pioneering German Palatine Doub (family), active in Western Maryland before and after American independence from England.  These pietist Doubs were early and active members of the Church of the United Brethren in Christ.  This conservative Trinitarian church claims to be the first Christian denomination founded in America and from its early days took a strong stand against slavery.

Ferdinando Fairfax: Ferdinando Fairfax (born in 1766 at Shannon Hill, Jefferson County, Virginia (now West Virginia); died on 24 September 1820 at Mount Eagle, Fairfax County, Virginia) was a Virginia landowner and member of the prominent Fairfax family.

Occoquan River: The Occoquan River is a tributary of the Potomac River in Northern Virginia, where it serves as part of the boundary between Fairfax and Prince William counties.  The river is 24.7 mi long, and its watershed covers about 590 sqmi .  It is formed by the confluence of Broad Run and Cedar Run in Prince William County; Bull Run, which forms Prince William County's boundary with Loudoun and the northerly part of Fairfax counties, enters it east-southeast of Manassas, as the Occoquan turns to the southeast.  It reaches the Potomac at Belmont Bay.  The Occoquan River is part of the Chesapeake Bay watershed.  The name "Occoquan" is derived from a Doeg Algonquian word translated as "at the end of the water".

James Fairfax: James Oswald Fairfax {'1': ", '2': ", '3': ", '4': "} (27 March 1933 – 11 January 2017) was an Australian company director, philanthropist, and a member of the Fairfax family, an Australian family prominent in the newspaper publisher industry.

Fairfax County Water Authority: Fairfax County Water Authority (FCWA or more recently Fairfax Water for short) is the main water company in the Northern Virginia region of the United States, and one of the three major water providers in the Washington, D.C. metropolitan area (the other two being the Washington Aqueduct and the Washington Suburban Sanitary Commission).  Fairfax Water serves most of the populated areas of Fairfax County, and also serves neighboring communities of Alexandria, Prince William County, Virginia and Loudoun County, Virginia through wholesale agreements.  It serves drinking water to 1.5 million people.

Vaucluse (plantation): Vaucluse was a villa in Fairfax County, Virginia, three miles (5 km) from Alexandria and 10 mi from Washington, D.C., on a hill near the Virginia Theological Seminary, that was owned first by Dr. James Craik, and later by the Fairfax family, the first being Thomas Fairfax, 9th Lord Fairfax of Cameron.

Catherine, Duchess of Cambridge: Catherine, Duchess of Cambridge (Catherine Elizabeth "Kate"; "née" Middleton; born 9 January 1982) is the wife of Prince William, Duke of Cambridge.  Following his father Charles, Prince of Wales, William is second in line to succeed his grandmother, Queen Elizabeth II, as monarch of the United Kingdom and 15 other Commonwealth realms, making Catherine a likely future queen consort.

Belmont Bay: Belmont Bay is a body of water at the mouth of the Occoquan River between Fairfax and Prince William counties, Virginia.  The bay covers about 1500 acre .  The bay adjoins the Elizabeth Hartwell Mason Neck National Wildlife Refuge and Mason Neck State Park on the Fairfax County side and the Occoquan Bay National Wildlife Refuge in Prince William County.  The bay was named for the home, "Belmont," which was built circa 1730 overlooking the bay by Catesby Cocke, who was the clerk of the Prince William County court.  Belmont Bay is notable for sightings of bald eagles that nest and feed in the refuges and for the numerous Great Blue Herons.  Belmont Bay is also a popular destination for pleasure boats.  Summer weekends usually attract 40 to as many as 100 boats to this location.

Thomas Fairfax (Walton): Sir Thomas Fairfax (  1450 – 31 March 1505) was the first member of the Fairfax family to own Gilling Castle, near Gilling East, North Riding of Yorkshire, England.  He is a direct ancestor of both Prince William, Duke of Cambridge and The Duchess of Cambridge.

Supporting paragraphs: Thomas Fairfax (Walton) Catherine, Duchess of Cambridge
Answer: Gilling Castle