Is the sun directly in the center of earth's orbit?

Technically the Earth doesn't orbit around the Sun and the Moon doesn't orbit around the Earth.

The Earth and the Moon both orbit around the centre of mass of the Earth and Moon. This is called the Earth-Moon Barycentre or EMB. It is about #4,671km# from the centre of the Earth or about #1,700km# below the Earth's surface.

In a similar way the Sun and all of the planets and other bodies in the Solar System orbit about the centre of mass of the Solar System. This is called the Solar Centre Barycentre.

The location of this barycentre is constantly changing due to the relative positions of the planets. It is usually just below the Sun's surface and sometimes even above the Sun's surface. It's position can be anywhere from close to the centre of the Sun to over #1,000,000km# from the centre of the Sun if all of the planets were aligned on the same side of the Sun.

The graph shows the position of the Solar System Barycentre relative to the centre of the Sun from 1938 to 2051. The unit of distance is the solar radius #696,000km#. The graph was plotted using NASA's DE430 data by a program I just wrote. The change in position is quite staggering. Most of the effects are due to Jupiter and Saturn.

So, the true story is that both the Sun and the Earth-Moon Barycentre orbit around the Solar System Barycentre.

Also, the Earth's orbit is slightly elliptical, with the Earth orbiting around one of the foci. The orbit is not a true ellipse as it is constantly being deformed by the gravitational pull of the other planets.

The Earth's perihelion distance #d_p=147,100km# and its aphelion distance #d_a=152,100km#. The semi-major axis #a# is defined as #2a=d_p+d_a#. This gives a value of #a=149,600km#. The distance of the centre of he ellipse from the focus is #a-d_p=#2,500,000km.

So, the distance of the Sun from the centre of the Earth's orbit is #2,500,000+-1,000,000km# depending upon the positions of the other planets.