Draw polar gradients for arcs and circle in Cairo as no native function in the library handles it. Done by drawing portions of arcs with incremental color steps.

polar_gradient.c

/** 
 * Draws an arc with a B&W gradient following the arc path.
 *  cr: the Cairo object to draw to
 *  lw: the linewidth
 *  nb_steps: the number of segments/color increments in which the arc will be splitted
 *  x_center: the x coordinate of the center of the arc
 *  y_center: the y coordinate of the center of the arc
 *  radius: the radius of the arc
 *  angle_from: the angle from which the arc will begin
 *  angle_to: the angle at which the arc will end
 *  color_from: the B&W starting color of the gradient, between 0.0 and 1.0
 *  color_to: the B&W ending color of the gradient, between 0.0 and 1.0
 *
 *  nb_steps must be adjusted to have a smooth gradient rendering depending on the displayed size of the element
 *  Usage example: 
 *  gradient_arc(cr, 0.125, 16, 0.5, 0.5, 0.5, -M_PI / 3.0, M_PI + M_PI / 3.0, 0.3, 0.9);
 *  
 */

void cairo_gradient_arc(cairo_t *cr, double lw, int nb_steps, double x_center, double y_center, double radius,
                   double angle_from, double angle_to, double color_from, double color_to)
{
  cairo_set_line_width(cr, lw);

  double *portions = malloc((1 + nb_steps) * sizeof(double));

  // note: cairo angles seems to be shifted by M_PI relatively to the unit circle
  angle_from = angle_from + M_PI;
  angle_to = angle_to + M_PI;
  double step = (angle_to - angle_from) / nb_steps;
  for(int i = 0; i < nb_steps; i++) portions[i] = angle_from + i * step;
  portions[nb_steps] = angle_to;

  for(int i = 0; i < nb_steps; i++)
  {
    double color = color_from + i * (color_to - color_from) / nb_steps;
    cairo_set_source_rgb(cr, color, color, color);
    cairo_arc(cr, x_center, y_center, radius, portions[i], portions[i + 1]);
    cairo_stroke(cr);
  }
  free(portions);
}

Sample produced by
gradient_arc(cr, 0.125, 16, 0.5, 0.5, 0.5, -M_PI / 3.0, M_PI + M_PI / 3.0, 0.3, 0.9);
to create a button decoration.