Crthuky

I'm still trying to get the hang of tikz.
I'm trying to shade a calotte within a sphere.
The code

begintikzpicture[font = sansmath]
 coordinate (O) at (0,0);



 % ball background color
 shade[ball color = blue, opacity = 0.1] (0,0) circle [radius = 2cm];

 % cone
 beginscope
 defrx0.71% horizontal radius of the ellipse
 defry0.15% vertical radius of the ellipse
 defz0.725% distance from center of ellipse to origin

 path [name path = ellipse] (0,z) ellipse (rx and ry);
 path [name path = horizontal] (-rx,z-ry*ry/z)
 -- (rx,z-ry*ry/z);
 path [name intersections = of = ellipse and horizontal];


 endscope


 % ball
 draw (O) circle [radius=2cm];
 % label of ball center point
 filldraw (O) circle (1pt) node[below] $O$;

 % radius
 draw[densely dashed] (O) to [edge label = $1$] (-1.33,1.33) coordinate (x);
 draw[densely dashed] (O) -- (1.33,1.33) coordinate (y);

 % cut of ball surface
 draw[red, densely dashed] (-1.36,1.46) arc [start angle = 170, end angle = 10,
 x radius = 13.8mm, y radius = 3.6mm];
 draw[red] (-1.29,1.52) arc [start angle=-200, end angle = 20,
 x radius = 13.75mm, y radius = 3.15mm];

 % label of cut of ball surface
 draw (-1.2,2.2) -- (-0.23,1.1) node at (-1.37,2.37) $B$;

 %angle
 pic[draw,->,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"][blue] angle=y--O--x;

endtikzpicture

gives me the sphere I'm working with and I want to have that:

I've tried fiddling with shade to no success.

Like this?

documentclass[tikz,border=3.14mm]standalone
usepackagesansmath
usepackagetikz-3dplot
usetikzlibrary3d,backgrounds,quotes,angles,calc,patterns
begindocument
tdplotsetmaincoords8000
begintikzpicture[tdplot_main_coords,font=sansmath]
 pgfmathsetmacroR2 % radius
 pgfmathsetmacromyang50 % latitude angle of the red circle
 coordinate (O) at (0,0,0);
 shade[ball color = blue, opacity = 0.2,tdplot_screen_coords] 
 (O) circle [radius = R*1cm];
 beginscope[canvas is xy plane at z=R*sin(myang),transform shape]
 % angVis from https://tex.stackexchange.com/a/49589/121799
 pgfmathsetmacroangVisatan(sin(myang)*cos(tdplotmaintheta)/sin(tdplotmaintheta))
 beginscope[on background layer]
 draw[red,dashed] (angVis:R*cos(myang)) arc (angVis:180-angVis:R*cos(myang));
 endscope 
 draw[red] (180-angVis:R*cos(myang)) arc (180-angVis:360+angVis:R*cos(myang));
 path (0:R*cos(myang)) coordinate (R) 
 (180:R*cos(myang)) coordinate (L);
 endscope
 beginscope[on background layer]
 draw[dashed] (L) -- (O) node[midway,below] $L$ -- (R);
 fill (O) circle[radius=1pt] node[below] $O$;
 pic[draw,-latex,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"] 
 angle=R--O--L;
 endscope
 path[tdplot_screen_coords,pattern=north east lines,pattern color=green!70!black] 
 let p1=(L),p2=(R),n1=atan2(y1,x1),n2=atan2(y2,x2)
 in (O) -- (R) arc(n2:n1:R) -- cycle;
endtikzpicture
enddocument

You can, of course, also add this to your code (which I find a bit more complicated than necessary, but this is only my opinion).

documentclass[tikz,border=3.14mm]standalone
usepackagesansmath
usetikzlibraryquotes,angles,%intersections,%<- not used
calc,patterns
begindocument

begintikzpicture[font = sansmath]
 coordinate (O) at (0,0);



 % ball background color
 shade[ball color = blue, opacity = 0.1] (0,0) circle [radius = 2cm];

 % cone you are not using this anywhere
% beginscope
% defrx0.71% horizontal radius of the ellipse
% defry0.15% vertical radius of the ellipse
% defz0.725% distance from center of ellipse to origin
% 
% path [name path global=ellipse] (0,z) ellipse (rx and ry);
% path [name path = horizontal] (-rx,z-ry*ry/z)
% -- (rx,z-ry*ry/z);
% 
% 
% 
% endscope


 % ball
 draw (O) circle [radius=2cm];
 % label of ball center point
 filldraw (O) circle (1pt) node[below] $O$;
 % radius
 draw[densely dashed] (O) to [edge label = $1$] (-1.33,1.33) coordinate (x);
 path[fill=green!70!black,fill opacity=0.5] 
 (O) -- (45:2) arc(45:135:2cm) -- cycle;
 draw[densely dashed] (O) -- (1.33,1.33) coordinate (y);

 % cut of ball surface
 draw[red, densely dashed] (-1.36,1.46) arc [start angle = 170, end angle = 10,
 x radius = 13.8mm, y radius = 3.6mm];
 draw[red] (-1.29,1.52) arc [start angle=-200, end angle = 20,
 x radius = 13.75mm, y radius = 3.15mm];

 % label of cut of ball surface
 draw (-1.2,2.2) -- (-0.23,1.1) node at (-1.37,2.37) $B$;

 %angle
 pic[draw,->,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"][blue] angle=y--O--x;
endtikzpicture
enddocument