Earth Gravity
Dave Typinski, 2008

Density Profile of the Earth - Spherical Approximation Using PREM Data

(* Earth's equatorial radius in meters- WGS 84 reference ellipsoid via Wikipedia *)
(Subscript[r, equatorial])=6.378137×(10^6); 

(* inverse flattening (1/f) - WGS 84 reference ellipsoid via Wikipedia *)
f=(1/298.257223563);

(Subscript[r, polar])=(Subscript[r, equatorial]) (1-f);

(Subscript[r, volumetric])=((((Subscript[r, equatorial])^2) (Subscript[r, polar]))^(1/3));

(* Quadratic and linear best fits to Preliminary Reference Earth Model, 1981 *)
(* Note that the PREM uses (Subscript[r, volumetric]) as the radius of the Earth *)
(* Input is in m, output is in kg·(m^(-3)) *)
(Subscript[h, innercore])=1.2215×(10^6); 
ρ[r_]:=-2.177275×(10^(-10)) (r^2)+1.911026×(10^(-8)) r+1.308848×(10^4)/;0≤r≤ (Subscript[h, innercore]);
(Subscript[h, outercore])=3.4800×(10^6);
ρ[r_]:=-2.412312×(10^(-10)) (r^2)+0.1397596×(10^(-3)) r+1.234623×(10^4)/;(Subscript[h, innercore])<r≤(Subscript[h, outercore]);
(Subscript[h, DLayer]) = 3.6300×(10^6);
ρ[r_]:=-5.000714×(10^(-4)) r+7306.695/;(Subscript[h, outercore])≤r<(Subscript[h, DLayer]);
(Subscript[h, lowermantle])=5.7010×(10^6);
ρ[r_]:=-3.092201×(10^(-11)) (r^2)-2.444117×(10^(-4)) r+6.782253×(10^3)/;(Subscript[h, DLayer])<r≤(Subscript[h, lowermantle]);
(Subscript[h, innertransitionzone1])=5.7710×(10^6);
ρ[r_]:=-2.328571×(10^(-4)) r+5.319659×(10^3)/;(Subscript[h, lowermantle])<r≤(Subscript[h, innertransitionzone1]);
(Subscript[h, innertransitionzone2])=5.9710×(10^6);
ρ[r_]:=-1.260300×(10^(-3)) r+1.124903×(10^4)/;(Subscript[h, innertransitionzone1])<r≤(Subscript[h, innertransitionzone2]);
(Subscript[h, outertransitionzone])=6.1510×(10^6);
ρ[r_]:=-5.970556×(10^(-4)) r+7.108267×(10^3)/;(Subscript[h, innertransitionzone2])<r≤(Subscript[h, outertransitionzone]);
(Subscript[h, lvlid])=6.3466×(10^6);
ρ[r_]:=1.086873×(10^(-4)) r+2.690961×(10^3)/;(Subscript[h, outertransitionzone])<r≤(Subscript[h, lvlid]);
(Subscript[h, innercrust])=6.3560×(10^6);
ρ[r_]:=2.900×(10^3)/;(Subscript[h, lvlid])<r≤(Subscript[h, innercrust]);
(Subscript[h, outercrust])=6.3680×(10^6);
ρ[r_]:=2.600×(10^3)/;(Subscript[h, innercrust])<r≤(Subscript[h, outercrust]);
(Subscript[h, ocean])=(Subscript[r, volumetric]);
ρ[r_]:=1.020×(10^3)/;(Subscript[h, outercrust])<r≤(Subscript[h, ocean]);

Plot[{ρ[x (10^3)] (10^(-3))},{x,0,(Subscript[h, ocean]) (10^(-3))},PlotRange->Automatic,PlotStyle->{{RGBColor[0,0,.7],Thin}},
Axes->None,GridLines->None,Frame->True,RotateLabel-> True,FrameStyle->{{Directive[Thin,FontFamily->Helvetica,14],Thin},{Directive[Thin,FontFamily->Helvetica,14],Thin}},FrameLabel->{Style["Radius, km",FontSize->14,FontFamily->Helvetica],Style["Density, kg·(m^(-3))×(10^3)",FontSize->14,FontFamily->Helvetica],Style["Earth's Density as a Function of Distance From Center",Bold,FontSize->16,FontFamily->Helvetica],""}]

Mass of the Earth - Spherical Approximation

(* Earth's equatorial radius in meters- WGS 84 reference ellipsoid via Wikipedia *)
(Subscript[r, equatorial])=6.378137×(10^6); 

(* inverse flattening (1/f) - WGS 84 reference ellipsoid via Wikipedia *)
f=(1/298.257223563);

(Subscript[r, polar])=(Subscript[r, equatorial]) (1-f);

(Subscript[r, volumetric])=((((Subscript[r, equatorial])^2) (Subscript[r, polar]))^(1/3));

(* Earth mass, NASA, kg *)
(Subscript[Mass, accepted])=5.9736×(10^24) ;

(* Quadratic fit to Preliminary Reference Earth Model (PREM), Dziewonski & Anderson, 1981 *)
(* Note that the PREM uses (Subscript[r, v]) as the radius of the Earth *)
(* Input is in m, output is in kg·(m^(-3)) *)
(Subscript[h, innercore])=1.2215×(10^6); 
ρ[r_]:=-2.177275×(10^(-10)) (r^2)+1.911026×(10^(-8)) r+1.308848×(10^4)/;0≤r≤ (Subscript[h, innercore]);
(Subscript[h, outercore])=3.4800×(10^6);
ρ[r_]:=-2.412312×(10^(-10)) (r^2)+0.1397596×(10^(-3)) r+1.234623×(10^4)/;(Subscript[h, innercore])<r≤(Subscript[h, outercore]);
(Subscript[h, DLayer]) = 3.6300×(10^6);
ρ[r_]:=-5.000714×(10^(-4)) r+7306.695/;(Subscript[h, outercore])≤r<(Subscript[h, DLayer]);
(Subscript[h, lowermantle])=5.7010×(10^6);
ρ[r_]:=-3.092201×(10^(-11)) (r^2)-2.444117×(10^(-4)) r+6.782253×(10^3)/;(Subscript[h, DLayer])<r≤(Subscript[h, lowermantle]);
(Subscript[h, innertransitionzone1])=5.7710×(10^6);
ρ[r_]:=-2.328571×(10^(-4)) r+5.319659×(10^3)/;(Subscript[h, lowermantle])<r≤(Subscript[h, innertransitionzone1]);
(Subscript[h, innertransitionzone2])=5.9710×(10^6);
ρ[r_]:=-1.260300×(10^(-3)) r+1.124903×(10^4)/;(Subscript[h, innertransitionzone1])<r≤(Subscript[h, innertransitionzone2]);
(Subscript[h, outertransitionzone])=6.1510×(10^6);
ρ[r_]:=-5.970556×(10^(-4)) r+7.108267×(10^3)/;(Subscript[h, innertransitionzone2])<r≤(Subscript[h, outertransitionzone]);
(Subscript[h, lvlid])=6.3466×(10^6);
ρ[r_]:=1.086873×(10^(-4)) r+2.690961×(10^3)/;(Subscript[h, outertransitionzone])<r≤(Subscript[h, lvlid]);
(Subscript[h, innercrust])=6.3560×(10^6);
ρ[r_]:=2.900×(10^3)/;(Subscript[h, lvlid])<r≤(Subscript[h, innercrust]);
(Subscript[h, outercrust])=6.3680×(10^6);
ρ[r_]:=2.600×(10^3)/;(Subscript[h, innercrust])<r≤(Subscript[h, outercrust]);
(Subscript[h, ocean])=(Subscript[r, volumetric]);
ρ[r_]:=1.020×(10^3)/;(Subscript[h, outercrust])<r≤(Subscript[h, ocean]);

Mass[a_,b_]:=(∫_0^(2 π))(∫_0^π)(∫_a^b)ρ[r] (r^2) Sin[ϕ]\[DifferentialD]r\[DifferentialD]ϕ\[DifferentialD]θ ;

(Subscript[Mass, partial])[1]=Mass[0,(Subscript[h, innercore])];
(Subscript[Mass, partial])[2]=Mass[(Subscript[h, innercore]),(Subscript[h, outercore])];
(Subscript[Mass, partial])[3]=Mass[(Subscript[h, outercore]),(Subscript[h, DLayer])];
(Subscript[Mass, partial])[4]=Mass[(Subscript[h, DLayer]),(Subscript[h, lowermantle])];
(Subscript[Mass, partial])[5]=Mass[(Subscript[h, lowermantle]),(Subscript[h, innertransitionzone1])];
(Subscript[Mass, partial])[6]=Mass[(Subscript[h, innertransitionzone1]),(Subscript[h, innertransitionzone2])];
(Subscript[Mass, partial])[7]=Mass[(Subscript[h, innertransitionzone2]),(Subscript[h, outertransitionzone])];
(Subscript[Mass, partial])[8]=Mass[(Subscript[h, outertransitionzone]),(Subscript[h, lvlid])];
(Subscript[Mass, partial])[9]=Mass[(Subscript[h, lvlid]),(Subscript[h, innercrust])];
(Subscript[Mass, partial])[10]=Mass[(Subscript[h, innercrust]),(Subscript[h, outercrust])];
(Subscript[Mass, partial])[11]=Mass[(Subscript[h, outercrust]),(Subscript[h, ocean])];

Print["The mass of the inner core is ",(Subscript[Mass, partial])[1]//N," kg."] 
Print["The mass of the outer core is ",(Subscript[Mass, partial])[2]//N," kg."]
Print["The mass of the D'' layer is ",(Subscript[Mass, partial])[3]//N," kg."]
Print["The mass of the lower mantle is ",(Subscript[Mass, partial])[4]//N," kg."]
Print["The mass of the inner transition zone 1 is ",(Subscript[Mass, partial])[5]//N," kg."]
Print["The mass of the inner transition zone 2 is ",(Subscript[Mass, partial])[6]//N," kg."]
Print["The mass of the outer transition zone is ",(Subscript[Mass, partial])[7]//N," kg."]
Print["The mass of the LV & Lid is ",(Subscript[Mass, partial])[8]//N," kg."]
Print["The mass of the inner crust is ",(Subscript[Mass, partial])[9]//N," kg."]
Print["The mass of the outer crust is ",(Subscript[Mass, partial])[10]//N," kg."]
Print["The mass of the ocean is ",(Subscript[Mass, partial])[11]//N," kg."]

(Subscript[Mass, total])=(∑_(i=1)^11)(Subscript[Mass, partial])[i]//N;
Print[""]
Print["The total mass of the Earth is ",(Subscript[Mass, total])," kg via integration of each layer."]
Print["The total mass of the Earth is ",(Subscript[Mass, accepted])," kg via NASA."]
Print[""]
Print["The error via integration of the PREM approximations is ",PaddedForm[100 (((Subscript[Mass, total]) - (Subscript[Mass, accepted]))/(Subscript[Mass, accepted]))//N,3]," percent."]
Print[""]

(Subscript[M, g])[r_]:= Mass[0,r]/;0≤r≤ (Subscript[h, innercore]);
(Subscript[M, g])[r_]:=((Subscript[Mass, partial])[1]+Mass[(Subscript[h, innercore]),r])/;(Subscript[h, innercore])<r≤ (Subscript[h, outercore]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,2}]+Mass[(Subscript[h, outercore]),r])/;(Subscript[h, outercore])<r≤ (Subscript[h, DLayer]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,3}]+Mass[(Subscript[h, DLayer]),r])/;(Subscript[h, DLayer])<r≤ (Subscript[h, lowermantle]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,4}]+Mass[(Subscript[h, lowermantle]),r])/;(Subscript[h, lowermantle])<r≤ (Subscript[h, innertransitionzone1]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,5}]+Mass[(Subscript[h, innertransitionzone1]),r])/;(Subscript[h, innertransitionzone1])<r≤ (Subscript[h, innertransitionzone2]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,6}]+Mass[(Subscript[h, innertransitionzone2]),r])/;(Subscript[h, innertransitionzone2])<r≤ (Subscript[h, outertransitionzone]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,7}]+Mass[(Subscript[h, outertransitionzone]),r])/;(Subscript[h, outertransitionzone])<r≤ (Subscript[h, lvlid]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,8}]+Mass[(Subscript[h, lvlid]),r])/;(Subscript[h, lvlid])<r≤ (Subscript[h, innercrust]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,9}]+Mass[(Subscript[h, innercrust]),r])/;(Subscript[h, innercrust])<r≤ (Subscript[h, outercrust]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,10}]+Mass[(Subscript[h, outercrust]),r])/;(Subscript[h, outercrust])<r≤ (Subscript[h, ocean]);
(Subscript[M, g])[r_]:=(Sum[(Subscript[Mass, partial])[i],{i,1,11}])/;(Subscript[h, ocean])<r;

ρavg=((3 (Subscript[M, g])[(Subscript[h, ocean])])/(4 π ((Subscript[h, ocean])^3)))//N;

Print["The average density of the Earth is ",ρavg," kg·(m^(-3))."]
Print[""]

(Subscript[M, prox])[r_]:=ρavg ((4 π (r^3))/3);

Plot[{(Subscript[M, prox])[x (10^3)],(Subscript[M, g])[x (10^3)]},{x,0,(Subscript[h, ocean]) (10^(-3))},PlotRange->Automatic,PlotStyle->{{Gray,Dashed,Thin},{RGBColor[0,0,.7],Thin}},
Axes->None,GridLines->None,Frame->True,RotateLabel-> True,FrameStyle->{{Directive[Thin,FontFamily->Helvetica,14],Thin},{Directive[Thin,FontFamily->Helvetica,14],Thin}},FrameLabel->{Style["Radius, km",FontSize->14,FontFamily->Helvetica],Style["Mass, kg",FontSize->14,FontFamily->Helvetica],Style["Earth's Mass \"Below\" Radius r as a Function of Distance From Center",Bold,FontSize->16,FontFamily->Helvetica],""}]

Acceleration due to Earth Gravity - Spherical Approximation

(* Universal gravitational constant, (m^3)·(kg^(-1))·(s^(-2)) *)
G=6.67428×(10^(-11));

(* Earth's equatorial radius in meters- WGS 84 reference ellipsoid via Wikipedia *)
(Subscript[r, equatorial])=6.378137×(10^6); 

(* inverse flattening (1/f) - WGS 84 reference ellipsoid via Wikipedia *)
f=(1/298.257223563);

(Subscript[r, polar])=(Subscript[r, equatorial]) (1-f);

(Subscript[r, volumetric])=((((Subscript[r, equatorial])^2) (Subscript[r, polar]))^(1/3));

Clear[ρ];

(* Quadratic fit to Preliminary Reference Earth Model (PREM), Dziewonski & Anderson, 1981 *)
(* Note that the PREM uses (Subscript[r, volumetric]) as the radius of the Earth *)
(* Input is in m, output is in kg·(m^(-3)) *)
(Subscript[h, innercore])=1.2215×(10^6); 
ρ[r_]:=-2.177275×(10^(-10)) (r^2)+1.911026×(10^(-8)) r+1.308848×(10^4)/;0≤r≤ (Subscript[h, innercore]);
(Subscript[h, outercore])=3.4800×(10^6);
ρ[r_]:=-2.412312×(10^(-10)) (r^2)+0.1397596×(10^(-3)) r+1.234623×(10^4)/;(Subscript[h, innercore])<r≤(Subscript[h, outercore]);
(Subscript[h, DLayer]) = 3.6300×(10^6);
ρ[r_]:=-5.000714×(10^(-4)) r+7306.695/;(Subscript[h, outercore])≤r<(Subscript[h, DLayer]);
(Subscript[h, lowermantle])=5.7010×(10^6);
ρ[r_]:=-3.092201×(10^(-11)) (r^2)-2.444117×(10^(-4)) r+6.782253×(10^3)/;(Subscript[h, DLayer])<r≤(Subscript[h, lowermantle]);
(Subscript[h, innertransitionzone1])=5.7710×(10^6);
ρ[r_]:=-2.328571×(10^(-4)) r+5.319659×(10^3)/;(Subscript[h, lowermantle])<r≤(Subscript[h, innertransitionzone1]);
(Subscript[h, innertransitionzone2])=5.9710×(10^6);
ρ[r_]:=-1.260300×(10^(-3)) r+1.124903×(10^4)/;(Subscript[h, innertransitionzone1])<r≤(Subscript[h, innertransitionzone2]);
(Subscript[h, outertransitionzone])=6.1510×(10^6);
ρ[r_]:=-5.970556×(10^(-4)) r+7.108267×(10^3)/;(Subscript[h, innertransitionzone2])<r≤(Subscript[h, outertransitionzone]);
(Subscript[h, lvlid])=6.3466×(10^6);
ρ[r_]:=1.086873×(10^(-4)) r+2.690961×(10^3)/;(Subscript[h, outertransitionzone])<r≤(Subscript[h, lvlid]);
(Subscript[h, innercrust])=6.3560×(10^6);
ρ[r_]:=2.900×(10^3)/;(Subscript[h, lvlid])<r≤(Subscript[h, innercrust]);
(Subscript[h, outercrust])=6.3680×(10^6);
ρ[r_]:=2.600×(10^3)/;(Subscript[h, innercrust])<r≤(Subscript[h, outercrust]);
(Subscript[h, ocean])=(Subscript[r, volumetric]);
ρ[r_]:=1.020×(10^3)/;(Subscript[h, outercrust])<r≤(Subscript[h, ocean]);

Mass[a_,b_]:=(∫_0^(2 π))(∫_0^π)(∫_a^b)ρ[r] (r^2) Sin[ϕ]\[DifferentialD]r\[DifferentialD]ϕ\[DifferentialD]θ ;

(Subscript[Mass, partial])[1]=Mass[0,(Subscript[h, innercore])];
(Subscript[Mass, partial])[2]=Mass[(Subscript[h, innercore]),(Subscript[h, outercore])];
(Subscript[Mass, partial])[3]=Mass[(Subscript[h, outercore]),(Subscript[h, DLayer])];
(Subscript[Mass, partial])[4]=Mass[(Subscript[h, DLayer]),(Subscript[h, lowermantle])];
(Subscript[Mass, partial])[5]=Mass[(Subscript[h, lowermantle]),(Subscript[h, innertransitionzone1])];
(Subscript[Mass, partial])[6]=Mass[(Subscript[h, innertransitionzone1]),(Subscript[h, innertransitionzone2])];
(Subscript[Mass, partial])[7]=Mass[(Subscript[h, innertransitionzone2]),(Subscript[h, outertransitionzone])];
(Subscript[Mass, partial])[8]=Mass[(Subscript[h, outertransitionzone]),(Subscript[h, lvlid])];
(Subscript[Mass, partial])[9]=Mass[(Subscript[h, lvlid]),(Subscript[h, innercrust])];
(Subscript[Mass, partial])[10]=Mass[(Subscript[h, innercrust]),(Subscript[h, outercrust])];
(Subscript[Mass, partial])[11]=Mass[(Subscript[h, outercrust]),(Subscript[h, ocean])];

(* Gravitational acceleration as a function of radius *)
(* Input is in m, output is in m·(s^(-2)) *)
(Subscript[a, g])[r_]:=(G/(r^2)) Mass[0,r]/;0≤r≤ (Subscript[h, innercore]);
(Subscript[a, g])[r_]:=(G/(r^2)) ((Subscript[Mass, partial])[1]+Mass[(Subscript[h, innercore]),r])/;(Subscript[h, innercore])<r≤ (Subscript[h, outercore]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,2}]+Mass[(Subscript[h, outercore]),r])/;(Subscript[h, outercore])<r≤ (Subscript[h, DLayer]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,3}]+Mass[(Subscript[h, DLayer]),r])/;(Subscript[h, DLayer])<r≤ (Subscript[h, lowermantle]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,4}]+Mass[(Subscript[h, lowermantle]),r])/;(Subscript[h, lowermantle])<r≤ (Subscript[h, innertransitionzone1]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,5}]+Mass[(Subscript[h, innertransitionzone1]),r])/;(Subscript[h, innertransitionzone1])<r≤ (Subscript[h, innertransitionzone2]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,6}]+Mass[(Subscript[h, innertransitionzone2]),r])/;(Subscript[h, innertransitionzone2])<r≤ (Subscript[h, outertransitionzone]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,7}]+Mass[(Subscript[h, outertransitionzone]),r])/;(Subscript[h, outertransitionzone])<r≤ (Subscript[h, lvlid]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,8}]+Mass[(Subscript[h, lvlid]),r])/;(Subscript[h, lvlid])<r≤ (Subscript[h, innercrust]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,9}]+Mass[(Subscript[h, innercrust]),r])/;(Subscript[h, innercrust])<r≤ (Subscript[h, outercrust]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,10}]+Mass[(Subscript[h, outercrust]),r])/;(Subscript[h, outercrust])<r≤ (Subscript[h, ocean]);
(Subscript[a, g])[r_]:=(G/(r^2)) (Sum[(Subscript[Mass, partial])[i],{i,1,11}])/;(Subscript[h, ocean])<r;

(* Gravitational acceleration as a function of radius, constant density *)
(* Input is in m, output is in m·(s^(-2)) *)
(Subscript[a, gconst])[r_]:=((G Sum[(Subscript[Mass, partial])[i],{i,1,11}] r)/((Subscript[h, ocean])^3)) /;0≤r≤ (Subscript[h, ocean]);
(Subscript[a, gconst])[r_]:=((G Sum[(Subscript[Mass, partial])[i],{i,1,11}])/(r^2)) /;(Subscript[h, ocean])<r;

(Subscript[a, WGS84avg])=9.8063;

ErrorPct=((N[(Subscript[a, g])[(Subscript[r, volumetric])]]-(Subscript[a, WGS84avg]))/(Subscript[a, WGS84avg])) 100

Print["The calculated acceleration due to gravity at Earth's volumetric radius is ",(Subscript[a, g])[(Subscript[r, volumetric])]//N," m·(s^(-2))."]
Print[""]
Print["The average surface acceleration using the WGS84 gravity formula is ",(Subscript[a, WGS84avg])," m·(s^(-2))."]
Print[""]
Print["The The relative error in the calculated acceleration is ",ErrorPct,"%."]


Plot[{(Subscript[a, g])[x (Subscript[r, volumetric])],(Subscript[a, gconst])[x (Subscript[r, volumetric])]},{x,0,2},PlotRange->{0,12},PlotStyle->{{RGBColor[0,0,.7],Thin},{Gray,Dashed,Thin}},Axes->None,GridLines->None,Frame->True,RotateLabel-> True,FrameStyle->{{Directive[Thin,FontFamily->Helvetica,14],Thin},{Directive[Thin,FontFamily->Helvetica,14],Thin}},FrameLabel->{Style["Earth Radii",FontSize->14,FontFamily->Helvetica],Style["Acceleration, m·(s^(-2))",FontSize->14,FontFamily->Helvetica],Style["Acceleration Due to Earth Gravity as a Function of Distance From Center",Bold,FontSize->16,FontFamily->Helvetica],""}]