Sargent was brought up in Burntisland, Fife, and schooled at Burntisland Primary School and Kirkcaldy High School. She completed a BSc Honours degree in Physics at the University of Edinburgh in 1963, and then immigrated to the United States, first studying at the University of California, Berkeley, and then from 1967 the California Institute of Technology, where she was awarded her Ph.D. She is currently the Ira S. Bowen Professor of Astronomy at Caltech and has served as director of the Owens Valley Radio Observatory and Combined Array for Research in Millimeter-wave Astronomy. She served as president of the American Astronomical Society from 2000 to 2002, continuing to serve on the council since. Sargent was the Vice President for Student Affairs at Caltech from December 1, 2007 until 2016..
Sargent was nominated in 2011 by President Obama to serve a six-year term on the National Science Board. She has served on committees such as the NRC's Committee for Astronomy and Astrophysics, NSF's Mathematical and Physical Sciences Advisory Committee, and in 1995/6 chaired the Visiting Committee to the National Radio Astronomy Observatory. She has been Chair of NASA's Space Science Advisory Committee since 1994. She is also Director of the Combined Array for Research in Millimeterwave Astronomy (CARMA).
Sargent won both the NASA Public Service Medal and the Caltech Woman of the Year Award in 1998. Asteroid 18244 Anneila is named in her honor. The University of Edinburgh named her Alumnus of the Year in 2002 and conferred an honorary degree of Doctor of Science on her in 2008. Sargent was elected an Honorary Fellow of the Royal Society of Edinburgh in 2017.
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