Do the rich pay enough taxes?

Walter Williams has this discussion from the Manhattan Institute:

According to IRS 2007 data, the richest 1 percent of Americans earned 22 percent of national personal income but paid 40 percent of all personal income taxes. The top 5 percent earned 37 percent and paid 61 percent of personal income tax. The top 10 percent earned 48 percent and paid 71 percent of all personal income taxes. The bottom 50 percent earned 12 percent of personal income but paid just 3 percent of income tax revenues. 

Some argue that these observations are misleading because there are other federal taxes the bottom 50 percenters pay such as Social Security and excise taxes. Moore presents data from the Tax Policy Center, run by the liberal Urban Institute and the Brookings Institution, that takes into account payroll and income taxes paid by different income groups. Because of the earned income tax credit, most of America's poor pay little or nothing. What the Tax Policy Center calls working class pay 3 percent of all federal taxes, middle class 11 percent, upper middle class 19 percent and wealthy 67 percent. 

President Obama and the Democratic Party harp about tax fairness. Here's my fairness question to you: What standard of fairness dictates that the top 10 percent of income earners pay 71 percent of the federal income tax burden while 47 percent of Americans pay absolutely nothing? . . .

Some great income tax data is available from the Tax Foundation here.  Using this data it would be possible to show how the top tax rate has been associated with the top 10%'s share of taxes from 1980 to 2009.  See related discussions by the Heritage Foundation.